The Database of Integer Sequences, Part 10 

     Part of the On-Line Encyclopedia of Integer Sequences


This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

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    home page:          www.research.att.com/~njas/
    


(start)



%I A078812
%S A078812 1,2,1,3,4,1,4,10,6,1,5,20,21,8,1,6,35,56,36,10,1,7,56,126,120,55,12,
%T A078812 1,8,84,252,330,220,78,14,1,9,120,462,792,715,364,105,16,1,10,165,792,
%U A078812 1716,2002,1365,560,136,18,1,11,220,1287,3432,5005,4368,2380,816,171,20
%N A078812 Triangle read by rows: T(n,k) = binomial(n+k-1,2*k-1).
%C A078812 Apart from signs, identical to A053122.
%C A078812 Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004
%C A078812 T(n,k)=number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2)=10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
%C A078812 T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%D A078812 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72 (2006), 51-62. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%H A078812 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b078812.txt">Rows n=0..50 of triangle, flattened</a>
%F A078812 G.f.: x*y/(1-(2+y)*x+x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.
%F A078812 If indexing begins at 0 we have: T(n, k) = (n+k+1)!/((n-k)!*(2k+1))!. T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n<k. T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n, k) = 0 if k<0, T(0, 0)=1 and T(0, k) = 0 for k>0. G.f. for the column k : Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2). Row sums : Sum_{k>=0} T(n, k) = A001906(n+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004
%F A078812 Diagonal sums are A000079(n)=sum{k=0..floor(n/2), binomial(n+k+1, n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004
%F A078812 Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), Oct 22 2006
%e A078812 Triangle begins:
%e A078812 .........................1
%e A078812 ........................2,.1
%e A078812 ......................3,.4,.1
%e A078812 ....................4,.10,.6,.1
%e A078812 ..................5,.20,.21,.8,.1
%e A078812 ................6,.35,.56,.36,.10,.1
%e A078812 .............7,.56,.126,.120,.55,.12,.1
%e A078812 ..........8,.84,.252,.330,.220,.78,.14,.1
%p A078812 for n from 1 to 11 do seq(binomial(n+k-1,2*k-1),k=1..n) od; # yields sequence in triangular form (Deutsch)
%o A078812 (PARI) T(n,k)=if(n<0,0,binomial(n+k-1,2*k-1))
%o A078812 (PARI) {T(n, k)=polcoeff( polcoeff( x*y/(1-(2+y)*x+x^2) +x*O(x^n), n), k)}
%Y A078812 This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.
%Y A078812 The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.
%Y A078812 Cf. A053123, A049310. Row sums give A001906.
%Y A078812 With signs: A053122.
%Y A078812 Adjacent sequences: A078809 A078810 A078811 this_sequence A078813 A078814 A078815
%Y A078812 Sequence in context: A137614 A143326 A053122 this_sequence A104711 A133112 A137649
%K A078812 easy,nice,nonn,tabl,new
%O A078812 0,2
%A A078812 Michael Somos, Dec 05, 2002
%E A078812 Edited by njas, Apr 28 2008

%I A104711
%S A104711 1,2,1,3,4,1,4,10,7,1,5,20,27,11,1,6,35,77,61,16,1,7,56,182,236,121,22,
%T A104711 1,8,84,378,726,611,218,29,1
%N A104711 Triangle, row sums = partial sums of Catalan numbers.
%C A104711 Row sums 1, 3, 8, 22, 64, 196...= partial sums of Catalan numbers, A014138. A104710 is extracted from A * B.
%F A104711 B = [1; 1, 1;, 1, 1, 1;...]; A = A001263, the Narayana triangle: [1; 1, 1; 1, 3, 1; 1, 6, 6, 1;...]. Consider both as infinite lower triangular matrices with the rest of the terms zeros. Extract the triangle A104711 from the product B * A.
%e A104711 First few rows of the triangle are:
%e A104711 1;
%e A104711 2, 1;
%e A104711 3, 4, 1;
%e A104711 4, 10, 7, 1;
%e A104711 5, 20, 27, 11, 1;
%e A104711 6, 35, 77, 61, 16, 1;
%e A104711 ...
%Y A104711 Cf. A014138, A104710, A005585, A000124.
%Y A104711 Adjacent sequences: A104708 A104709 A104710 this_sequence A104712 A104713 A104714
%Y A104711 Sequence in context: A143326 A053122 A078812 this_sequence A133112 A137649 A104002
%K A104711 nonn,tabl,uned
%O A104711 1,2
%A A104711 Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2005

%I A133112
%S A133112 1,2,1,3,4,1,4,10,8,1,5,20,35,16,1,6,35,112,126,32,1,7,56,294,672,462,
%T A133112 64,1,8,84,672,2772,4224,1716,128,1,9,120,1386,9504,28314,27456,6435,
%U A133112 256,1,10,165,2640,28314,151008,306735,183040,24310,512,1
%N A133112 Triangular array, read by rows, associated with sums of certain Vandermonde determinants.
%C A133112 Appears to be the same as A073165 read as a triangular array (excluding the first column).
%F A133112 T(n,k)=1/(1!*2! ... *k!)*sum {1 <= x_1, ...,x_k <= n}|det V(x_1, ...,x_k)|, where V(x_1, ...,x_k} is the Vandermonde matrix of order k. For example, T(n,2) = 1/2*sum {1<=i,j<= n} |i-j|. T(n,k) = 1/(1!*2! ... *k!)*sum {1 <= x_1, ...,x_k <= n} |(product {1 <= i < j <= k} (x_j - x_i) )|.
%e A133112 Triangle starts
%e A133112 1;
%e A133112 2 1;
%e A133112 3 4 1;
%e A133112 4 10 8 1;
%e A133112 5 20 35 16 1;
%Y A133112 A000292 (column 2), A040977 (column 3), A133111 (column 4). Cf. A103905.
%Y A133112 Adjacent sequences: A133109 A133110 A133111 this_sequence A133113 A133114 A133115
%Y A133112 Sequence in context: A053122 A078812 A104711 this_sequence A137649 A104002 A073135
%K A133112 easy,nonn,tabl
%O A133112 1,2
%A A133112 Peter Bala (pbala(AT)toucansurf.com), Sep 18 2007

%I A137649
%S A137649 1,2,1,3,4,1,4,11,7,1,5,26,32,11,1,6,57,122,76,16,1,7,120,423,426,156,
%T A137649 22,1,8,247,1389,2127,1206,288,29,1,9,502,4414,9897,8157,2934,491,37,1,
%U A137649 10,1013,13744,44002,50682,25761,6371,787,46,1
%N A137649 Triangle read by rows, A000012 * A008277.
%C A137649 Row sums = A024716: (1, 3, 8, 23, 75, 278,...). A137650 = A008277 * A000012
%F A137649 A000012 * A008277 (Stirling2 triangle) as infinite lower triangular matrices. Partial column sums of A008277.
%e A137649 First few rows of the triangle are:
%e A137649 1;
%e A137649 2, 1;
%e A137649 3, 4, 1;
%e A137649 4, 11, 7, 1;
%e A137649 5, 26, 32, 11, 1;
%e A137649 6, 57, 122, 76, 16, 1;
%e A137649 7, 120, 423, 426, 156, 22, 1;
%e A137649 ...
%e A137649 Row 4 = (4, 11, 7, 1) = partial column sums of the first 4 rows of A008277:
%e A137649 1;
%e A137649 1, 1;
%e A137649 1, 3, 1;
%e A137649 1, 7, 6, 1;
%e A137649 ...
%Y A137649 Cf. A024716, A008277.
%Y A137649 Adjacent sequences: A137646 A137647 A137648 this_sequence A137650 A137651 A137652
%Y A137649 Sequence in context: A078812 A104711 A133112 this_sequence A104002 A073135 A063804
%K A137649 nonn,tabl
%O A137649 1,2
%A A137649 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 01 2008

%I A104002
%S A104002 1,2,1,3,4,1,4,12,6,1,5,32,27,8,1,6,80,108,48,10,1,7,192,405,256,75,12,
%T A104002 1,8,448,1458,1280,500,108,14,1,9,1024,5103,6144,3125,864,147,16,1,10,
%U A104002 2304,17496,28672,18750,6480,1372,192,18,1,11,5120,59049,131072
%N A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern, and containing it exactly once.
%C A104002 T(n+k,k+1) = total number of occurrences of any given letter in all possible n-length words on a k-letter alphabet. For example, with the 2 letter alphabet {0,1} there are 4 possible 2-length words: {00,01,10,11}. The letter 0 occurs 4 times altogether, as does the letter 1. T(4,3) = 4. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 03 2007
%H A104002 T. Mansour, <a href="http://arXiv.org/abs/math.CO/9911243">Permutations containing and avoiding certain patterns</a>
%F A104002 T(n, k) = (n-k+1) * (k-1)^(n-k), k<=n.
%e A104002 1
%e A104002 2,1
%e A104002 3,4,1
%e A104002 4,12,6,1
%e A104002 5,32,27,8,1
%e A104002 6,80,108,48,10,1
%e A104002 7,192,405,256,75,12,1
%e A104002 8,448,1458,1280,500,108,14,1
%Y A104002 Cf. Left-edge columns include A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, A053541, A081127, A081128.
%Y A104002 Adjacent sequences: A103999 A104000 A104001 this_sequence A104003 A104004 A104005
%Y A104002 Sequence in context: A104711 A133112 A137649 this_sequence A073135 A063804 A078753
%K A104002 nonn,tabl
%O A104002 2,2
%A A104002 Ralf Stephan, Feb 26 2005

%I A073135
%S A073135 1,2,1,3,4,1,4,12,6,1,6,32,27,8,1,10,81,108,48,10,1,17,200,406,256,75,
%T A073135 12,1,28,488,1470,1281,500,108,14,1,45,1184,5193,6160,3126,864,147,16,
%U A073135 1,72,2865,18036,28832,18770,6481,1372,192,18,1,116,6924,61885,132352
%N A073135 Table by antidiagonals of T(n,k)=2n*T(n,k-1)-n^2*T(n,k-2)+T(n,k-4) starting with T(n,1)=1.
%F A073135 T(n, k) =(A073133(n, k+2)-A073134(n, k+2))/2 =sum_j{0<=j<=[(k-1)/4]} abs(A053122(k-3j-1, j)*n^(k-4j-1))
%e A073135 Rows start: 1,2,3,4,6,10,17,...; 1,4,12,32,81,200,488,...; 1,6,27,108,406,1470,5193,...; 1,8,48,256,1281,6160,28832,...; etc.
%Y A073135 Rows include A024490, A048776. Columns include A000012, A005843, A033428, A033430.
%Y A073135 Adjacent sequences: A073132 A073133 A073134 this_sequence A073136 A073137 A073138
%Y A073135 Sequence in context: A133112 A137649 A104002 this_sequence A063804 A078753 A119443
%K A073135 nonn,tabl
%O A073135 0,2
%A A073135 Henry Bottomley (se16(AT)btinternet.com), Jul 16 2002

%I A063804
%S A063804 1,1,2,1,3,4,1,4,12,17,1,5,25,206,143,1,6,50,6029,181472,4890,1,7,
%T A063804 91,508321
%N A063804 Triangle T(n,k) (n >= 3, k = 1..n-2) giving number of nonisomorphic oriented matroids with n points in n-k dimensions.
%H A063804 L. Finschi, <a href="http://www.om.math.ethz.ch">Homepage of Oriented Matroids</a>
%H A063804 L. Finschi and K. Fukuda, <a href="http://compgeo.math.uwaterloo.ca/~cccg01/proceedings/short/finschi-1053.ps">Complete combinatorial generation of small point set configurations and hyperplane arrangements</a>, pp. 97-100 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%e A063804 1; 1,2; 1,3,4; 1,4,12,17; ...
%Y A063804 Diagonals give A063800-A063803. Row sums give A063805. For nondegenerate matroids see A063851.
%Y A063804 Adjacent sequences: A063801 A063802 A063803 this_sequence A063805 A063806 A063807
%Y A063804 Sequence in context: A137649 A104002 A073135 this_sequence A078753 A119443 A126198
%K A063804 nonn,tabl,nice
%O A063804 3,3
%A A063804 njas, Aug 20 2001

%I A078753
%S A078753 1,1,2,1,3,4,1,5,6,1,8,1,1,10,11,2,1,13,2,15,16,1,18,1,3,20,1,4,23,24,1,
%T A078753 1,26,5,28,29,2,1,32,1,33,2,7,36,1,8,3,40,1,41,42,1,44,45,10,47,4,1,2,1,
%U A078753 11,4,53,12,55,2,1,58,59,14,1,5,2,63,64,1,6,67,16,3,70,1,72,1,1,74,3,18
%N A078753 Number of steps to factor 2n+1 using Fermat's factorization method.
%C A078753 Smallest positive integer k such that (ceiling(sqrt(n))+k-1)^2 - n is a square.
%D A078753 Tattersall, J. "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, pp. 102-103.
%e A078753 To factor 931 using Fermat's method we need four iterations: 31^2 - 931 = 30, 32^2 - 931 = 93, 33^2 - 931 = 158, 34^2 - 931 = 225 = 15^2. Hence 931 = (34 - 15)(34 + 15)=19 * 49; so a(931)=4.
%t A078753 f[n_] := Module[{x0, x}, x0=Ceiling[Sqrt[n]]; For[x=x0, True, x++, If[IntegerQ[Sqrt[x^2-n]], Return[x-x0+1]]]]; f/@Range[3, 201, 2] g[n_] := Module[{d, r}, d=Divisors[n]; r=d[[Floor[Length[d]/2]+1]]; (r+n/r)/2-Ceiling[Sqrt[n]]+1]; g/@Range[3, 201, 2] (PARI): f(n, k) = (ceil(sqrt(n))+k-1)^2-n forstep(n=3, 201, 2, k=1; while(!issquare(f(n, k)), k++); print1(k", "))
%Y A078753 Adjacent sequences: A078750 A078751 A078752 this_sequence A078754 A078755 A078756
%Y A078753 Sequence in context: A104002 A073135 A063804 this_sequence A119443 A126198 A055888
%K A078753 easy,nonn
%O A078753 1,3
%A A078753 Jason Earls (zevi_35711(AT)yahoo.com), Dec 22 2002

%I A119443
%S A119443 1,2,1,3,4,1,5,6,4,6,1,7,10,12,9,12,8,1,11,14,20,9,15,36,8,12,24,10,1
%N A119443 Distribution of A060642 in Abramowitz and Stegun order having shape sequence A000041.
%H A119443 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%F A119443 a(n) = A119441(n) * A048996(n)
%Y A119443 Cf. A000041, A048996, A119441.
%Y A119443 Adjacent sequences: A119440 A119441 A119442 this_sequence A119444 A119445 A119446
%Y A119443 Sequence in context: A073135 A063804 A078753 this_sequence A126198 A055888 A094442
%K A119443 easy,more,nonn
%O A119443 1,2
%A A119443 Alford Arnold (Alford1940(AT)aol.com), May 22 2006

%I A126198
%S A126198 1,1,2,1,3,4,1,5,7,8,1,8,13,15,16,1,13,24,29,31,32,1,21,44,56,61,63,64,
%T A126198 1,34,81,108,120,125,127,128,1,55,149,208,236,248,253,255,256,1,89,274,
%U A126198 401,464,492,504,509,511,512,1,144,504,773,912,976,1004,1016,1021,1023,1024
%N A126198 Triangle read by rows: T(n,k) (1<=k<=n) = number of compositions of n into parts of size <= k.
%D A126198 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.
%F A126198 G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
%F A126198 T(n,3)=A008937(n)-A008937(n-3) for n>=3. T(n,4)=A107066(n-1)-A107066(n-5) for n>=5. T(n,5)=A001949(n+4)-A001949(n-1) for n>=5. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007
%e A126198 Triangle begins:
%e A126198 1
%e A126198 1 2
%e A126198 1 3 4
%e A126198 1 5 7 8
%e A126198 1 8 13 15 16
%e A126198 1 13 24 29 31 32
%e A126198 1 21 44 56 61 63 64
%e A126198 Could also be extended to a square array:
%e A126198 1 1 1 1 1 1 1 1 ...
%e A126198 1 2 2 2 2 2 2 2 ...
%e A126198 1 3 4 4 4 4 4 4 ...
%e A126198 1 5 7 8 8 8 8 8 ...
%e A126198 1 8 13 15 16 16 16 ...
%e A126198 1 13 24 29 31 32 32 ...
%e A126198 1 21 44 56 61 63 64 ...
%e A126198 which when read by antidiagonals (downwards) gives A048887.
%p A126198 A126198 := proc(n,k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)),x=0,n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ",A126198(n,k)); od; od; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007
%Y A126198 Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions.
%Y A126198 2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
%Y A126198 Cf. A082267.
%Y A126198 Adjacent sequences: A126195 A126196 A126197 this_sequence A126199 A126200 A126201
%Y A126198 Sequence in context: A063804 A078753 A119443 this_sequence A055888 A094442 A060642
%K A126198 nonn,tabl,nice
%O A126198 1,3
%A A126198 njas, Mar 09 2007
%E A126198 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007

%I A055888
%S A055888 1,1,2,1,3,4,1,5,8,8,1,6,16,20,16,1,8,25,46,48,32,1,9,37,84,124,112,
%T A055888 64,1,11,50,142,256,320,256,128,1,12,67,216,480,732,800,576,256,1,14,
%U A055888 84,319,812,1500,2000,1952,1280,512,1,15,105,443,1304,2772,4432,5280
%N A055888 Invert transform of partition triangle A008284.
%H A055888 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%e A055888 1; 1,2; 1,3,4; 1,5,8,8; 1,6,16,20,16; ...
%Y A055888 Row sums give A055887.
%Y A055888 Adjacent sequences: A055885 A055886 A055887 this_sequence A055889 A055890 A055891
%Y A055888 Sequence in context: A078753 A119443 A126198 this_sequence A094442 A060642 A049400
%K A055888 nonn,tabl
%O A055888 1,3
%A A055888 Christian G. Bower (bowerc(AT)usa.net), Jun 09 2000

%I A094442
%S A094442 1,2,1,3,4,1,5,9,6,1,8,20,18,8,1,13,40,50,30,10,1,21,78,120,100,45,12,1,
%T A094442 34,147,273,280,175,63,14,1,55,272,588,728,560,280,84,16,1,89,495,1224,
%U A094442 1764,1638,1008,420,108,18,1,144,890,2475,4080,4410,3276,1680,600,135
%N A094442 Triangular array T(n,k)=F(n+2-k)C(n,k), k=0,1,2,...,n; n>=0.
%e A094442 First four rows:
%e A094442 1
%e A094442 2 1
%e A094442 3 4 1
%e A094442 5 9 6 1
%Y A094442 Cf. A094437, A000045.
%Y A094442 Adjacent sequences: A094439 A094440 A094441 this_sequence A094443 A094444 A094445
%Y A094442 Sequence in context: A119443 A126198 A055888 this_sequence A060642 A049400 A106382
%K A094442 nonn,tabl
%O A094442 1,2
%A A094442 Clark Kimberling (ck6(AT)evansville.edu), May 03 2004

%I A060642
%S A060642 1,2,1,3,4,1,5,10,6,1,7,22,21,8,1,11,43,59,36,10,1,15,80,144,124,55,12,
%T A060642 1,22,141,321,362,225,78,14,1,30,240,669,944,765,370,105,16,1,42,397,
%U A060642 1323,2266,2287,1437,567,136,18,1,56,640,2511,5100,6215,4848,2478,824
%N A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.
%C A060642 Row sums give A055887
%F A060642 G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 02 2004
%e A060642 Table begins 1; 2 1; 3 4 1; 5 10 6 1; ...
%e A060642 For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.
%Y A060642 Cf. A000041, A048574, A055887, A055888.
%Y A060642 Adjacent sequences: A060639 A060640 A060641 this_sequence A060643 A060644 A060645
%Y A060642 Sequence in context: A126198 A055888 A094442 this_sequence A049400 A106382 A004741
%K A060642 easy,nonn,tabl
%O A060642 1,2
%A A060642 Alford Arnold (Alford1940(AT)aol.com), Apr 16 2001
%E A060642 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 02 2004

%I A049400
%S A049400 1,1,2,1,3,4,1,6,9,10,1,10,21,25,26,1,20,51,70,75,76,1,35,127,196,225,
%T A049400 231,232,1,70,323,588,715,756,763,764,1,126,835,1764,2347,2556,2611,2619,2620,1,252,2188,5544,7990,
%U A049400 9096,9415,9486,9495,9496,1,462,5798,17424,27908,33231,35135,35596
%N A049400 Partial sums of rows of A047884. Young Tableaux by height.
%H A049400 <a href="http://www.research.att.com/~njas/sequences/Sindx_Y.html#Young">Index entries for sequences related to Young tableaux.</a>
%e A049400 1; 1,2; 1,3,4; 1,6,9,10; ...
%Y A049400 Cf. A000085, A007579, A007578, A007580, A049401.
%Y A049400 Adjacent sequences: A049397 A049398 A049399 this_sequence A049401 A049402 A049403
%Y A049400 Sequence in context: A055888 A094442 A060642 this_sequence A106382 A004741 A133923
%K A049400 easy,nice,nonn,tabl
%O A049400 1,3
%A A049400 Alford Arnold (Alford1940(AT)AOL.com)

%I A106382
%S A106382 1,2,1,3,4,1,11,6,13,14,0,20,17,29
%N A106382 Imaginary parts of numbers defined in A106381.
%Y A106382 Cf. A106381.
%Y A106382 Adjacent sequences: A106379 A106380 A106381 this_sequence A106383 A106384 A106385
%Y A106382 Sequence in context: A094442 A060642 A049400 this_sequence A004741 A133923 A125158
%K A106382 nonn
%O A106382 1,2
%A A106382 Sven Simon (sven-h.simon(AT)t-online.de), Apr 30 2005

%I A004741
%S A004741 1,2,1,3,4,2,1,3,5,6,4,2,1,3,5,7,8,6,4,2,1,3,5,7,9,10,8,6,4,2,1,3,5,
%T A004741 7,9,10,8,6,4,2,1,3,5,7,9,11,12,10,8,6,4,2,1,3,5,7,9,11,13,14,12,10,8,6,4,2,1,
%U A004741 3,5,7,9,11,13,15,16,14,12,10,8,6,4,2
%N A004741 Concatenation of sequences (1,3,..,2n-1,2n,2n-2,..,2) for n >= 1.
%C A004741 Odd numbers increasing from 1 to 2k-1 followed by even numbers decreasing from 2k to 2.
%C A004741 Also called Smarandache Permutation Sequence.
%C A004741 The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.
%D A004741 J. Brown et al., Problem 4619, School Science and Mathematics (USA), Vol. 97(4), 1997, pp. 221-222.
%D A004741 F. Smarandache, "Numerical Sequences", University of Craiova, 1975; [Arizona State University, Special Collection, Tempe, AZ, USA].
%D A004741 F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse., Bucharest, 1996.
%D A004741 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
%H A004741 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/">Smarandache Notions Journal</a>
%H A004741 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/CP2.pdf">Collected Papers, Vol. II</a>
%H A004741 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheSequences.html">Link to a section of The World of Mathematics.</a>
%H A004741 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%F A004741 Ordinal transform of A004737. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006
%Y A004741 Adjacent sequences: A004738 A004739 A004740 this_sequence A004742 A004743 A004744
%Y A004741 Sequence in context: A060642 A049400 A106382 this_sequence A133923 A125158 A112384
%K A004741 nonn,easy
%O A004741 1,2
%A A004741 R. Muller

%I A133923
%S A133923 1,1,2,1,3,4,2,1,4,2,1,2,1,4,2,1,2,1,2,1,4,2,1,8,4,2,1,6,3,12,6,3,6,3,8,
%T A133923 4,2,1,2,1,2,1,2,1,6,3,4,2,1,6,3,12,6,3,8,4,2,1,2,1,2,6,3,16,8,4,2,1
%N A133923 2-divided number of divisors of n*a(n-1).
%C A133923 Let A, B, C are integers. We have a class of sequences a(n)= d(A*a(n-1)+B) if a(n-1) is not divisible by C, else a(n)= a(n-1)/C.
%F A133923 a(n)= d(n*a(n-1)) if a(n-1)is not divisible by 2 else a(n)= a(n-1)/2 . d(n*a(n-1)) is the number of divisors of n*a(n-1)
%Y A133923 Cf. A000005.
%Y A133923 Adjacent sequences: A133920 A133921 A133922 this_sequence A133924 A133925 A133926
%Y A133923 Sequence in context: A049400 A106382 A004741 this_sequence A125158 A112384 A123390
%K A133923 nonn,uned
%O A133923 1,3
%A A133923 Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), Jan 07 2008

%I A125158
%S A125158 1,1,2,1,3,4,2,1,5,3,6,4,7,2,8,1,9,5,10,11,3,6,12,13,4,7,14,2,15,16,8,1,
%T A125158 17,18,9,19,5,20,10,21,11,3,22,6,23,24,12,25,13,4,26,27,7,28,14,2,29,30,
%U A125158 15,31,16,32,8,1,33,34,17,35,18,36,9,37,19,38,5,39,20,40,10,41,21,42,11
%N A125158 The fractal sequence associated with A125150.
%C A125158 If you delete the first occurrence of each n, the remaining sequence is the original sequence; thus the sequence contains itself as a proper subsequence (infinitely many times).
%D A125158 C. Kimberling, "Interspersions and fractal sequences associated with fractions (c^j)/(d^k)," preprint, 2006.
%H A125158 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal Sequences</a>.
%F A125158 a(n)=number of the row of array A125150 that contains n.
%e A125158 1 is in row 1 of A125150; 2 in row 1; 3 in row 2;
%e A125158 4 in row 1; 5 in row 3; 6 in row 4, so the fractal
%e A125158 sequence starts with 1,1,2,1,3,4
%Y A125158 Cf. A125150.
%Y A125158 Adjacent sequences: A125155 A125156 A125157 this_sequence A125159 A125160 A125161
%Y A125158 Sequence in context: A106382 A004741 A133923 this_sequence A112384 A123390 A088208
%K A125158 nonn
%O A125158 1,3
%A A125158 Clark Kimberling (ck6(AT)evansville.edu), Nov 21 2006

%I A112384
%S A112384 1,1,2,1,3,4,2,1,5,3,6,7,8,4,2,1,9,10,11,12,5,3,6,7,13,14,8,4,15,2,
%T A112384 16,17,18,19,20,1,9,10,11,12,21,22,23,5,3,6,24,25,26,27,28,29,7,13,
%U A112384 14,8,4,15,30,31,32,33,34,35,36,2,16,17,18,19,20,1,37,38,39,40,41
%N A112384 A self-descriptive fractal sequence: the sequence contains every positive integer. If the first occurrence of each integer is deleted from the sequence, the resulting sequence is the same is the original (this process may be called "upper trimming").
%C A112384 This sequence describes the number of Xs that are dropped and the number of numbers written between dropped Xs (cf. A112382 and A112383).
%Y A112384 Cf. A112377, A112382, A112383.
%Y A112384 Adjacent sequences: A112381 A112382 A112383 this_sequence A112385 A112386 A112387
%Y A112384 Sequence in context: A004741 A133923 A125158 this_sequence A123390 A088208 A081878
%K A112384 nonn
%O A112384 0,3
%A A112384 Kerry Mitchell (lkmitch(AT)gmail.com), Dec 05 2005

%I A123390
%S A123390 1,2,1,3,4,2,1,5,6,3,7,8,4,2,1,9,10,5,11,12,6,3,13,14,7,15,16,8,4,2,1,
%T A123390 17,18,9,19,20,10,5,21,22,11,23,24,12,6,3,25,26,13,27,28,14,7,29,30,15,
%U A123390 31,32,16,8,4,2,1,33,34,17,35,36,18,9,37,38,19,39,40,20,10,5,41,42,21
%N A123390 Triangle read by rows: n-th row starts with n and continues with half the previous value as long as that is even.
%C A123390 A fractal sequence, generated by the rule a(n) is a new maximum when a(n-1) is odd, and a repetition of an earlier value when a(n-1) is even.
%F A123390 a(1) = 1, for n > 1, if a(n-1) is even, a(n) = a(n-1)/2, otherwise a(n) = (max_{k<n} a(k)) + 1.
%F A123390 Ordinal transform of A082850.
%Y A123390 Row lengths are A001511. Cf. A120385.
%Y A123390 Adjacent sequences: A123387 A123388 A123389 this_sequence A123391 A123392 A123393
%Y A123390 Sequence in context: A133923 A125158 A112384 this_sequence A088208 A081878 A088606
%K A123390 nonn,tabf
%O A123390 1,2
%A A123390 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 13 2006

%I A088208
%S A088208 1,1,2,1,3,4,2,1,5,7,3,4,8,6,2,1,9,13,5,7,15,11,3,4,12,16,8,6,14,10,2,1,
%T A088208 17,25,9,13,29,21,5,7,23,31,15,11,27,19,3,4,20,28,12,16,32,24,8,6,22,30,
%U A088208 14,10,26,18,2,1,33,49,17,25,57,41,9,13,45,61,29,21,53,37,5,7,39,55,23
%N A088208 Table read by rows where T(0,0)=1; n-th row has 2^n terms T(n,j),j=0 to 2^n-1. For j==0 mod 2, T(n+1,2j)=T(n,j) and T(n+1,2j+1)=T(n,j)+2^n. For j==1 mod 2, T(n+1,2j+1)=T(n,j) and T(n+1,2j)=T(n,j)+2^n.
%C A088208 Schroeder, p. 281 states "The ordering with which the iterates x_n fall into the 2^m different chaos bands [order as to magnitude] is also the same as the ordering of the iterates in a stable orbit of period length P = 2^m. For example, for both the period-4 orbit and the four chaos bands, the iterates, starting with the largest iterate x_1, are ordered as follows: x_1 > x_3 > x_4 > x_2."
%D A088208 Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W.H. Freeman, 1991, p. 282.
%e A088208 1
%e A088208 1 2
%e A088208 1 3 4 2
%e A088208 1 5 7 3 4 8 6 2
%e A088208 1 9 13 5 7 15 11 3 4 12 16 8 6 14 10 2
%Y A088208 Cf. A088372.
%Y A088208 Adjacent sequences: A088205 A088206 A088207 this_sequence A088209 A088210 A088211
%Y A088208 Sequence in context: A125158 A112384 A123390 this_sequence A081878 A088606 A140073
%K A088208 nonn,tabl
%O A088208 1,3
%A A088208 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 23 2003
%E A088208 Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net) and njas, Oct 08 2003

%I A081878
%S A081878 1,2,1,3,4,2,1,5,8,4,2,1,6,3,7,10,5,8,4,2,1,8,4,2,1,9,10,5,8,4,2,1,10,5,
%T A081878 8,4,2,1,11,14,7,10,5,8,4,2,1,12,6,3,13,16,8,4,2,1,14,7,10,5,8,4,2,1,15,
%U A081878 16,8,4,2,1,16,8,4,2,1,17,20,10,5,8,4,2,1,18,9,10,5,8,4,2,1,19,22,11,14
%N A081878 Triangle read by rows in which row n begins with n (n=1,2,3,...) and iterates the process of dividing n by 2 if n is even, adding 3 if n is an odd prime, otherwise adding 1; stopping when either 1 or 3 is reached.
%C A081878 The number of steps to reach 1 or 3 is in A081879.
%C A081878 The sequence is well defined: iteration of f (the map defining the sequence) terminates either at 1 or 3 for all values of n>0. Proof: Assuming that all natural numbers < k converge, then if k is even it converges (as f(k)=k/2 < k) and if it is odd, then f(f(k)) is either (k+1)/2 or (k+3)/2, and these are less than k for all k>4.
%e A081878 1; 2,1; 3; 4,2,1; 5,8,4,2,1; 6,3; 7,10,5,8,4,2,1; ...
%o A081878 (PARI) xnprp3(n) = { for(x=1,n, p1 = x; print1(x" "); while(p1>1, if(p1%2==0,p1/=2, if(isprime(p1),p1+=3,p1 = p1+1;)); print1(p1" "); if(p1==3,break) ) ) }
%o A081878 (MIT Scheme) (define (isprime? n) (cond ((< n 4) (> n 1)) (else (let loop ((i (floor->exact (/ n 2)))) (cond ((= 1 i) #t) ((zero? (modulo n i)) #f) (else (loop (-1+ i))))))))
%o A081878 (define (A081878 upto-n) (let outloop ((x 1) (a (list))) (cond ((> x upto-n) (reverse! a)) (else (let inloop ((a (cons x a))) (let ((n (car a))) (cond ((and (not (= 1 n)) (not (= 3 n))) (cond ((even? n) (inloop (cons (/ n 2) a))) ((isprime? n) (inloop (cons (+ n 3) a))) (else (inloop (cons (1+ n) a))))) (else (outloop (1+ x) a)))))))))
%Y A081878 Cf. A081879.
%Y A081878 Adjacent sequences: A081875 A081876 A081877 this_sequence A081879 A081880 A081881
%Y A081878 Sequence in context: A112384 A123390 A088208 this_sequence A088606 A140073 A131389
%K A081878 easy,nonn,tabf
%O A081878 1,2
%A A081878 Cino Hilliard (hillcino368(AT)gmail.com), Apr 12 2003
%E A081878 Edited by Antti Karttunen (his-firstname.his-surname(AT)iki.fi) and Jud McCranie (j.mccranie(AT)comcast.net), Jun 03, 2003

%I A088606
%S A088606 0,1,0,1,2,1,3,4,2,2,1,1,2,4,3,3,3,4,1,2,1,1,2,3,1,5,1,5,1,2,4,2,2,4,11,
%T A088606 1,4,1,3,6,2,1,3,1,5,6,4,1,5,1,5,2,4,2,3,6,2,3,1,2,1,2,1,2,3,6,3,4,2,4,
%U A088606 6,3,1,1,6,2,2,4,12,1,5,4,5,1,1,5,3,3,3,3,2,5,1,3,1,2,17,2,1,3,3,2,5,5
%N A088606 Smallest number k such that concatenation of k and prime(n) is a prime, or 0 if no other number exists. a(1) = a(3) = 0.
%C A088606 Subsidiary sequences: (set(1)) Index of the start of the first occurrence of a string of n consecutive 1's or 2's or 3's etc. (set (2)): a(n) = smallest prime such that concatenation of 1 with n successive primes starting from a(n) gives primes in each case. ( n primes are obtained.) Similarly for 2,3, etc. Conjecture: The subsidiary sequences are infinite.
%Y A088606 Cf. A096915.
%Y A088606 Adjacent sequences: A088603 A088604 A088605 this_sequence A088607 A088608 A088609
%Y A088606 Sequence in context: A123390 A088208 A081878 this_sequence A140073 A131389 A131394
%K A088606 base,nonn
%O A088606 1,5
%A A088606 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 15 2003
%E A088606 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 18 2003

%I A140073
%S A140073 1,2,1,3,4,2,3,1,6,6,3,5,7,6,7,8,9,9,5,2,10,4,6,7,9,11,9,5,6,8,12,4,7,3,
%T A140073 13,11,10,12,6,8,14,3,2,15,11,3,10,12,16,1,15,16,15,8,9,17,3,6,4,15,18,
%U A140073 18,14,16,18,18,15,6,5,11,19,6,14,20,12,19,13,3,21,21,15,1,9,19,6,20,9
%N A140073 Values of y in A033205.
%Y A140073 Cf. A033205, A140072.
%Y A140073 Adjacent sequences: A140070 A140071 A140072 this_sequence A140074 A140075 A140076
%Y A140073 Sequence in context: A088208 A081878 A088606 this_sequence A131389 A131394 A130585
%K A140073 nonn
%O A140073 1,2
%A A140073 Zak Seidov (zakseidov(AT)yahoo.com), May 05 2008

%I A131389
%S A131389 0,1,2,1,3,4,2,3,6,4,5,7,5,8,6,7,9,10,8,11,10,12,9,11,13,14,13,12,15,16,
%T A131389 14,15,17,18,16,19,17,20,19,21,18,22,20,21,24,22,23,25,23,26,25,27,24,
%U A131389 28,26,27,30,29,31,32,28,31,29,33,30,34,32,33,35,36,34
%V A131389 0,1,2,-1,3,4,-2,-3,6,-4,5,7,-5,8,-6,-7,9,10,-8,11,-10,12,-9,-11,13,14,-13,-12,15,16,
%W A131389 -14,-15,17,18,-16,19,-17,20,-19,21,-18,22,-20,-21,24,-22,23,25,-23,26,-25,27,-24,28,
%X A131389 -26,-27,30,29,-31,32,-28,31,-29,33,-30,34,-32,-33,35,36,-34
%N A131389 Conjectured permutation of the integers using Rule 1 with A131388(1)=1.
%C A131389 Rule 1 is given at A131388. Conjecture : A131389 is a permutation of the integers.
%F A131389 This is the sequence d( ) in the formula for A131388.
%e A131389 See A131388.
%Y A131389 Cf. A131388, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397.
%Y A131389 Adjacent sequences: A131386 A131387 A131388 this_sequence A131390 A131391 A131392
%Y A131389 Sequence in context: A081878 A088606 A140073 this_sequence A131394 A130585 A125100
%K A131389 sign
%O A131389 1,3
%A A131389 Clark Kimberling (ck6(AT)evansville.edu), Jul 05 2007

%I A131394
%S A131394 0,1,2,1,3,4,2,3,6,4,5,7,5,8,6,7,9,10,8,11,10,12,9,11,13,14,13,15,12,15,
%T A131394 17,16,14,17,18,19,16,20,19,21,23,22,18,23,20,21,24,22,25,26,24,27,26,
%U A131394 28,25,27,32,29,28,30,29,31,37,34,36,33,39,35,33
%V A131394 0,1,2,-1,3,4,-2,-3,6,-4,5,7,-5,8,-6,-7,9,10,-8,11,-10,12,-9,-11,13,14,-13,15,-12,-15,
%W A131394 17,16,-14,-17,18,19,-16,20,-19,21,-23,22,-18,23,-20,-21,24,-22,25,26,-24,27,-26,28,
%X A131394 -25,-27,32,29,-28,30,-29,31,-37,34,-36,33,-39,35,-33
%N A131394 Conjectured permutation of the integers using Rule 2 with A131393(1)=1.
%C A131394 Rule 2 is given at A131393. Conjecture : A131394 is a permutation of the integers.
%F A131394 This is the sequence d( ) in the formula for A131393.
%e A131394 See A131393.
%Y A131394 Cf. A131388, A131389, A131390, A131391, A131392, A131393, A131395, A131396, A131397.
%Y A131394 Adjacent sequences: A131391 A131392 A131393 this_sequence A131395 A131396 A131397
%Y A131394 Sequence in context: A088606 A140073 A131389 this_sequence A130585 A125100 A128544
%K A131394 sign
%O A131394 1,3
%A A131394 Clark Kimberling (ck6(AT)evansville.edu), Jul 05 2007

%I A130585
%S A130585 1,2,1,3,4,2,4,7,6,2,5,16,24,16,4,6,15,22,20,10,2,7,36,90,120,90,36,6,8,
%T A130585 35,90,142,140,84,28,4,9,52,170,336,420,336,168,48,6,10,53,168,352,508,
%U A130585 504,336,144,36,4
%N A130585 A054522 * A007318.
%C A130585 A130584 = A007318 * A054522 Row sums = A130586: (1, 3, 9, 19, 65, 75, 385,...).
%F A130585 A054522 * A007318 as infinite lower triangular matrices.
%e A130585 First few rows of the triangle are:
%e A130585 1;
%e A130585 2, 1;
%e A130585 3, 4, 2;
%e A130585 4, 7, 6, 2;
%e A130585 5, 16, 24, 16, 4;
%e A130585 6, 15, 22, 20, 10, 2;
%e A130585 7, 36, 90, 120, 90, 36, 6;
%e A130585 ...
%Y A130585 Cf. A054522, A007318, A130585, A130586.
%Y A130585 Adjacent sequences: A130582 A130583 A130584 this_sequence A130586 A130587 A130588
%Y A130585 Sequence in context: A140073 A131389 A131394 this_sequence A125100 A128544 A120058
%K A130585 nonn,tabl
%O A130585 0,2
%A A130585 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2007

%I A125100
%S A125100 1,2,1,3,4,2,4,9,9,3,5,16,24,16,5,6,25,50,50,30,8,7,36,90,120,105,54,13,
%T A125100 8,49,147,245,280,210,98,21,9,64,224,448,630,616,420,176,34,10,81,324,
%U A125100 756,1260,1512,1344,828,315,55,11,100,450,1200,2310,3276,3570,2880,1620
%N A125100 Triangle read by rows: T(n,k)=fibonacci(k+1)*binom(n,k)+(k+1)*binom(n,k+1) (0<=k<=n).
%C A125100 A081663: (1, 3, 9, 25, 66, 169...) is the binomial transform of A081659, (n + F(n+1).
%C A125100 Binomial transform of the bidiagonal matrix with the Fibonacci numbers (1,1,2,3,5,8...) in the main diagonal and (1,2,3...) in the subdiagonal. Sum of terms in row n = n*2^(n-1)+fibonacci(2n+1) (A081663).
%e A125100 First few rows of the triangle are:
%e A125100 1;
%e A125100 2, 1;
%e A125100 3, 4, 2;
%e A125100 4, 9, 9, 3;
%e A125100 5, 16, 24, 16, 5;
%e A125100 6, 25, 50, 50, 30, 8;
%e A125100 7, 36, 90, 120, 105, 54, 13;
%e A125100 8, 49, 147, 245, 280, 210, 98, 21;
%e A125100 ...
%p A125100 with(combinat): T:=(n,k)->binomial(n,k)*fibonacci(k+1)+(k+1)*binomial(n,k+1): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A125100 Cf. A081663, A081659.
%Y A125100 Cf. A000045.
%Y A125100 Adjacent sequences: A125097 A125098 A125099 this_sequence A125101 A125102 A125103
%Y A125100 Sequence in context: A131389 A131394 A130585 this_sequence A128544 A120058 A102756
%K A125100 nonn,tabl
%O A125100 0,2
%A A125100 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 20 2006
%E A125100 Edited by njas, Nov 29 2006

%I A128544
%S A128544 1,2,1,3,4,2,4,9,9,3,5,16,24,17,5,6,25,50,55,33,8,7,36,90,135,123,61,13,
%T A128544 8,49,147,280,343,259,112,21,9,64,224,518,798,812,532,202,34,10,81,324,
%U A128544 882,1638,2100,1848,1062,361,55
%N A128544 A007318 * A128540.
%C A128544 Row sums = A069403: (1, 3, 9, 25, 67,...).
%F A128544 A007318 * A128540 as infinite lower triangular matrices; (binomial transform of A128540).
%e A128544 First few rows of the triangle are:
%e A128544 1;
%e A128544 2, 1;
%e A128544 3, 4, 2;
%e A128544 4, 9, 9, 3;
%e A128544 5, 16, 24, 17, 5;
%e A128544 6, 25, 50, 55, 33, 8;
%e A128544 7, 36, 90, 135, 123, 61, 13;
%e A128544 ...
%Y A128544 Cf. A007318, A128540, A069403.
%Y A128544 Adjacent sequences: A128541 A128542 A128543 this_sequence A128545 A128546 A128547
%Y A128544 Sequence in context: A131394 A130585 A125100 this_sequence A120058 A102756 A086614
%K A128544 nonn,tabl
%O A128544 0,2
%A A128544 Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 10 2007

%I A120058
%S A120058 1,2,1,3,4,2,4,9,10,4,5,16,28,24,8,6,25,60,80,56,16,7,36,110,200,216,
%T A120058 128,32,8,49,182,420,616,560,288,64,9,64,280,784,1456,1792,1408,640,128,
%U A120058 10,81,408,1344,3024,4704,4992,3456,1408,256
%V A120058 1,2,-1,3,-4,2,4,-9,10,-4,5,-16,28,-24,8,6,-25,60,-80,56,-16,7,-36,110,-200,216,-128,
%W A120058 32,8,-49,182,-420,616,-560,288,-64,9,-64,280,-784,1456,-1792,1408,-640,128,10,-81,408,
%X A120058 -1344,3024,-4704,4992,-3456,1408,-256
%N A120058 Coefficients for obtaining A120057 from Bell numbers.
%C A120058 Appears to be essentially the same as A056863, but (as of Jun 06 2006) that sequence definition is unclear and there are discrepencies in the signs.
%C A120058 Alternating column sums appear to be 3^n.
%F A120058 A120057(n,k) = sum_{i=1,k} T(n,i)*B(n-i+1).
%F A120058 T(n,k) = Sum_j A120095(n,j) * S1(j,n-k+1), where S1 is the Stirling numbers of the first kind (A008275).
%F A120058 Unsigned version, as an infinite lower triangular matrix, equals A007318 * A134315. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 19 2007
%e A120058 Table starts:
%e A120058 1
%e A120058 2,-1
%e A120058 3,-4,2
%e A120058 4,-9,10,-4
%e A120058 5,-16,28,-24,8
%e A120058 6,-25,60,-80,56,-16
%Y A120058 Cf A120057, A000110, A056863.
%Y A120058 Cf. A008275, A120095.
%Y A120058 Cf. A134315.
%Y A120058 Adjacent sequences: A120055 A120056 A120057 this_sequence A120059 A120060 A120061
%Y A120058 Sequence in context: A130585 A125100 A128544 this_sequence A102756 A086614 A108959
%K A120058 sign,tabl
%O A120058 1,2
%A A120058 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 06 2006, Jun 07 2006

%I A102756
%S A102756 1,2,1,3,4,2,4,10,10,3,5,20,31,20,5,6,35,76,78,40,8,7,56,161,232,184,76,
%T A102756 13,8,84,308,582,636,406,142,21,9,120,546,1296,1831,1604,861,260,34,10,
%U A102756 165,912,2640,4630,5215,3820,1766,470,55
%N A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by : T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-2,k-2)-T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if n<k.
%C A102756 Rising and falling diagonals are A008999, A124400.
%F A102756 Sum_{k, 0<=k<=n}x^k*T(n,k)=A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for n=-1, 0, 1, 2, 3 . Sum_{k, 0<=k<=n}(-2)^k*T(n,k)= [1,0,3,0,9,0,27,0,81,0,...]for n=0,1,2,3,4,5,..., powers of 3 alternating with zeros . T(n,n-1)=2*A001629(n+1) for n>=1 . T(n,n)=Fibonacci(n+1)=A000045(n+1) . T(n,0)=n+1 . T(n,1)=A000292(n) for n>=1 . T(n+1,2)= binomial(n+4,n-1)+binomial(n+2,n-1)=A051747(n) for n>=1.
%e A102756 Triangle begins:
%e A102756 1;
%e A102756 2, 1;
%e A102756 3, 4, 2;
%e A102756 4, 10, 10, 3;
%e A102756 5, 20, 31, 20, 3;
%e A102756 6, 35, 76, 78, 40, 8;
%e A102756 7, 56, 161, 232, 184, 76, 13;
%e A102756 8, 84, 308, 582, 636, 406, 142, 21;
%e A102756 9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
%e A102756 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
%Y A102756 Adjacent sequences: A102753 A102754 A102755 this_sequence A102757 A102758 A102759
%Y A102756 Sequence in context: A125100 A128544 A120058 this_sequence A086614 A108959 A107893
%K A102756 nonn,tabl
%O A102756 0,2
%A A102756 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 18 2006

%I A086614
%S A086614 1,2,1,3,4,2,4,10,12,5,5,20,42,40,14,6,35,112,180,140,42,7,56,252,600,
%T A086614 770,504,132,8,84,504,1650,3080,3276,1848,429,9,120,924,3960,10010,
%U A086614 15288,13860,6864,1430,10,165,1584,8580,28028,57330,73920,58344,25740
%N A086614 Triangle of coefficients, read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.
%e A086614 Rows:
%e A086614 {1},
%e A086614 {2,1},
%e A086614 {3,4,2},
%e A086614 {4,10,12,5},
%e A086614 {5,20,42,40,14},
%e A086614 {6,35,112,180,140,42},
%e A086614 {7,56,252,600,770,504,132},
%e A086614 {8,84,504,1650,3080,3276,1848,429}, ...
%Y A086614 Cf. A086615 (antidiagonal sums), A086616 (row sums), A086617.
%Y A086614 Adjacent sequences: A086611 A086612 A086613 this_sequence A086615 A086616 A086617
%Y A086614 Sequence in context: A128544 A120058 A102756 this_sequence A108959 A107893 A131987
%K A086614 nonn,tabl
%O A086614 0,2
%A A086614 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 24 2003

%I A108959
%S A108959 1,2,1,3,4,2,4,10,14,7,5,20,54,76,38,6,35,154,419,590,295,7,56,364,1616,
%T A108959 4400,6196,3098,8,84,756,4962,22048,60036,84542,42271,9,120
%N A108959 Triangle arising in connection with deformations of type D Kleinian singularities.
%D A108959 P. Boddington, Ph.D. thesis, University of Warwick, 2005.
%F A108959 For k>=0 define p_k(x)=x(x+1)(x+3)...(x+k(k-1)/2) and consider the linear map taking each p_k(x) to kp_k(x)/x. Then the images of x, x^2, x^3, ... are given by the rows. Eg x^3 goes to 3x^2+4x+2.
%Y A108959 This sequence is an improved version of A097418. Coefficients of 1 give A000366.
%Y A108959 Adjacent sequences: A108956 A108957 A108958 this_sequence A108960 A108961 A108962
%Y A108959 Sequence in context: A120058 A102756 A086614 this_sequence A107893 A131987 A120874
%K A108959 easy,nonn,tabl
%O A108959 0,2
%A A108959 Paul Boddington (psb(AT)maths.warwick.ac.uk), Jul 22 2005

%I A107893
%S A107893 1,2,1,3,4,2,4,11,14,6,5,26,64,66,24,6,57,244,456,384,120,7,120,846,
%T A107893 2556,3744,2640,720,8,247,2778,12762,28944,34560,20880,5040
%N A107893 Triangle read by rows, related to A055129 (repunits in base k).
%C A107893 Second column of A107893 = Eulerian numbers (A000295) starting with 1: 1, 4, 11, 26, 57... Rightmost term in row n = (n-1)!
%F A107893 n-th row = inverse binomial transform of n-th column of A055129; where the latter are generated from f(x) = x^(n-1) + x^(n-2) + ...+ x + 1; (x=1, 2, 3...)
%e A107893 Binomial transform of Row 4 in the form: (4, 11, 14, 6, 0, 0, 0...) = Row 4 of A055129: 4, 15, 40, 85, ...which is generated from f(x) = x^3 + x^2 + x + 1; (x=1,2,3...).
%Y A107893 Cf. A055129, A000295.
%Y A107893 Adjacent sequences: A107890 A107891 A107892 this_sequence A107894 A107895 A107896
%Y A107893 Sequence in context: A102756 A086614 A108959 this_sequence A131987 A120874 A112382
%K A107893 nonn,tabl
%O A107893 1,2
%A A107893 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 26 2005

%I A131987
%S A131987 1,2,1,3,4,2,5,1,6,3,7,8,4,9,2,10,5,11,1,12,6,13,3,14,7,15,16,8,17,4,18,
%T A131987 9,19,2,20,10,21,5,22,11,23,1,24,12,25,6,26,13,27,3,28,14,29,7,30,15,31,
%U A131987 32,16,33,8,34,17,35,4,36,18,37,9,38,19,39,2,40,20,41,10,42,21,43
%N A131987 Representation of a dense para-sequence.
%C A131987 A fractal sequence. The para-sequence may be regarded as a sort of "limit" of the concatenated segments. The para-sequence (itself not a sequence) is dense in the sense that every pair of terms i and j are separated by another term (and hence separated by infinitely many terms).
%C A131987 The para-sequence accounts for positions of dyadic rational numbers in the following way: Label 1/2 as 1; label 1/4, 3/4 as 2 and 3; label 1/8, 3/8, 5/8, 7/8 as 4,5,6,7, etc. Then, for example, the ordering 1/8 < 1/4 < 3/8 < 1/2 < 5/8 < 3/4 < 7/8 matches the labels 4,2,5,1,6,3,7, which is the 3rd segment of A131987. The n-th segment consists of labels for rationals having denominators 2, 4, 8, ..., 2^n.
%D A131987 C. Kimberling, Proper self-containing sequences, fractal sequences, and para-sequences, preprint, 2007.
%F A131987 Start with 1 and isolate it using 2,3 like this: 2,1,3. Then isolate those using 4,5,6,7, like this: 4,2,5,1,6,3,7. Continue, and concatenate.
%e A131987 The next segment, to be concatenated after 4,2,5,1,6,3,7, is
%e A131987 8,4,9,2,10,5,11,1,12,6,13,3,14,7,15.
%Y A131987 Cf. A133108.
%Y A131987 Adjacent sequences: A131984 A131985 A131986 this_sequence A131988 A131989 A131990
%Y A131987 Sequence in context: A086614 A108959 A107893 this_sequence A120874 A112382 A117384
%K A131987 nonn
%O A131987 1,2
%A A131987 Clark Kimberling (ck6(AT)evansville.edu), Aug 05 2007, Sep 12 2007

%I A120874
%S A120874 1,2,1,3,4,2,5,1,6,7,3,8,9,4,10,2,11,12,5,13,1,14,15,6,16,17,7,18,3,19,
%T A120874 20,8,21,22,9,23,4,24,25,10,26,2,27,28,11,29,30,12,31,5,32,33,13,34,1,
%U A120874 35,36,14,37,38,15,39,6,40,41,16,42,43,17,44,7,45,46,18,47,3,48,49,19
%N A120874 Fractal sequence of the Fraenkel array (A038150).
%C A120874 A fractal sequence f contains itself as a proper subsequence; e.g., if you delete the first occurrence of each positive integer, the remaining sequence is f; thus f properly contains itself infinitely many times.
%D A120874 C. Kimberling, The equation (j+k+1)^2-4k=Q*n^2 and related dispersions, preprint.
%H A120874 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/classic.html#WYTH">Classic Sequences</a>.
%e A120874 The fractal sequence f(n) of a dispersion D={d(g,h,)} is defined
%e A120874 as follows. For each positive integer n there is a unique (g,h)
%e A120874 such that n=d(g,h), and f(n)=g. So f(6)=2 because the row of
%e A120874 the Fraenkel array in which 6 occurs is row 2.
%Y A120874 Cf. A038150.
%Y A120874 Adjacent sequences: A120871 A120872 A120873 this_sequence A120875 A120876 A120877
%Y A120874 Sequence in context: A108959 A107893 A131987 this_sequence A112382 A117384 A125160
%K A120874 nonn
%O A120874 1,2
%A A120874 Clark Kimberling (ck6(AT)evansville.edu), Jul 10 2006

%I A112382
%S A112382 1,1,2,1,3,4,2,5,1,6,7,8,3,9,10,11,12,4,13,14,2,15,16,17,18,19,5,20,
%T A112382 1,21,22,23,24,25,26,6,27,28,29,30,31,32,33,7,34,35,36,37,38,39,40,
%U A112382 41,8,42,43,44,3,45,46,47,48,49,50,51,52,53,9,54,55,56,57,58,59,60
%N A112382 A self-descriptive fractal sequence: the sequence contains every positive integer. If the first occurrence of each integer is deleted from the sequence, the resulting sequence is the same is the original (this process may be called "upper trimming").
%C A112382 This sequence is also self-descriptive in that each element gives the number of first occurrences of integers (X's in the example) that were removed just before it.
%e A112382 If we denote the first occurrence of each integer by X we get:
%e A112382 X, 1, X, 1, X, X, 2, X, 1, X, X, X, 3, X, X, X, X, 4, X, X, 2, ...
%e A112382 and dropping the X's:
%e A112382 1, 1, 2, 1, 3, 4, 2, ...
%e A112382 which is the beginning of the original sequence.
%Y A112382 Cf. A112377, A112383, A112384.
%Y A112382 Adjacent sequences: A112379 A112380 A112381 this_sequence A112383 A112384 A112385
%Y A112382 Sequence in context: A107893 A131987 A120874 this_sequence A117384 A125160 A009947
%K A112382 nonn
%O A112382 0,3
%A A112382 Kerry Mitchell (lkmitch(AT)gmail.com), Dec 05 2005

%I A117384
%S A117384 1,2,1,3,4,2,5,3,6,7,4,8,5,9,6,10,11,7,12,8,13,9,14,10,15,16,11,17,12,
%T A117384 18,13,19,14,20,15,21,22,16,23,17,24,18,25,19,26,20,27,21,28,29,22,30,
%U A117384 23,31,24,32,25,33,26,34,27,35,28,36,37,29,38,30,39,31,40,32,41,33,42
%N A117384 Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 4*a(n), starting with a(1)=1, and filling the next vacant position with the smallest unused number.
%C A117384 Positions where n occurs are A001614(n) and 4*n-A001614(n), where A001614 is the Connell sequence: 1 odd, 2 even, 3 odd, ...
%F A117384 a(4*a(n)-n) = a(n). Limit a(n)/n = 1/2. Limit (a(n+1)-a(n))/sqrt(n) = 1.
%F A117384 a( A001614(n) ) = n; a( 4n - A001614(n) ) = n.
%e A117384 9 first appears at position: A001614(9) = 14;
%e A117384 9 next appears at position: 4*9 - A001614(9) = 22.
%o A117384 (PARI) {a(n)=local(A=vector(n),m=1); for(k=1,n,if(A[k]==0,A[k]=m;if(4*m-k<=#A,A[4*m-k]=m);m+=1));A[n]}
%Y A117384 Cf. A117385 (a(5*a(n)-n)=a(n)), A117386 (a(6*a(n)-n)=a(n)).
%Y A117384 Cf. A001614 (Connell sequence).
%Y A117384 Adjacent sequences: A117381 A117382 A117383 this_sequence A117385 A117386 A117387
%Y A117384 Sequence in context: A131987 A120874 A112382 this_sequence A125160 A009947 A026249
%K A117384 nonn
%O A117384 1,2
%A A117384 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2006

%I A125160
%S A125160 1,2,1,3,4,2,5,6,1,7,8,9,3,10,4,11,12,13,14,2,15,5,16,17,6,18,1,19,20,7,
%T A125160 21,22,23,8,24,25,26,9,27,3,28,29,10,30,4,31,32,11,33,34,12,35,36,13,37,
%U A125160 38,14,39,40,2,41,42,43,15,44,45,46,5,16,47,48,17
%N A125160 The fractal sequence associated with A125152.
%C A125160 If you delete the first occurrence of each n, the remaining sequence is the original sequence; thus the sequence contains itself as a proper subsequence (infinitely many times).
%D A125160 C. Kimberling, "Interspersions and fractal sequences associated with fractions (c^j)/(d^k)," preprint, 2006.
%H A125160 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal Sequences</a>.
%F A125160 a(n)=number of the row of array A125152 that contains n.
%e A125160 1 is in row 1 of A125152; 2 in row 2; 3 in row 1;
%e A125160 4 in row 3; 5 in row 4; 6 in row 2, so the fractal
%e A125160 sequence starts with 1,2,1,3,4,2
%Y A125160 Cf. A125152.
%Y A125160 Adjacent sequences: A125157 A125158 A125159 this_sequence A125161 A125162 A125163
%Y A125160 Sequence in context: A120874 A112382 A117384 this_sequence A009947 A026249 A130527
%K A125160 nonn
%O A125160 1,2
%A A125160 Clark Kimberling (ck6(AT)evansville.edu), Nov 21 2006

%I A009947
%S A009947 0,0,1,2,1,3,4,2,5,6,3,7,8,4,9,10,5,11,12,6,13,14,7,15,16,
%T A009947 8,17,18,9,19,20,10,21,22,11,23,24,12,25,26,13,27,28,14,
%U A009947 29,30,15,31,32,16,33,34,17,35,36,18,37,38,19,39,40
%N A009947 Sequence of nonnegative integers, but insert n/2 after every even number n.
%C A009947 Coefficients in expansion of e/3 = Sum a(n)/n!, n=1..inf, using greedy algorithm.
%C A009947 Numerators of Peirce sequence of order 2.
%D A009947 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 151.
%p A009947 A009947 := proc(a,n) local i,b,c; b := a; c := [ floor(b) ]; for i from 1 to n-1 do b := b-c[ i ]/i!; c := [ op(c), floor(b*(i+1)!) ]; od; c; end:
%Y A009947 Cf. A071281.
%Y A009947 Adjacent sequences: A009944 A009945 A009946 this_sequence A009948 A009949 A009950
%Y A009947 Sequence in context: A112382 A117384 A125160 this_sequence A026249 A130527 A026366
%K A009947 nonn,easy
%O A009947 0,4
%A A009947 R. W. Gosper

%I A026249
%S A026249 1,2,1,3,4,2,5,6,7,3,8,9,4,10,11,12,5,13,14,6,15,16,7,17,18,19,
%T A026249 8,20,21,9,22,23,24,10,25,26,11,27,28,12,29,30,31,13,32,33,14,
%U A026249 34,35,36,15,37,38,16,39,40,41,17,42,43,18,44,45,19,46,47,48
%N A026249 a(n) = j if n = [ j*sqrt(2) ], else a(n) = k if n = [ k*(2 + sqrt(2)) ].
%Y A026249 Adjacent sequences: A026246 A026247 A026248 this_sequence A026250 A026251 A026252
%Y A026249 Sequence in context: A117384 A125160 A009947 this_sequence A130527 A026366 A122164
%K A026249 nonn
%O A026249 1,2
%A A026249 Clark Kimberling (ck6(AT)evansville.edu)

%I A130527
%S A130527 0,1,2,1,3,4,2,5,6,7,3,8,9,4,10,11,12,5,13,14,6,15,16,7,17,18,19,8,20,
%T A130527 21,9,22,23,24,10,25,26,11,27,28,12,29,30,31,13,32,33,14,34,35,36,15,37,
%U A130527 38,16,39,40,41,17,42,43,18,44,45,19,46,47,48
%V A130527 0,1,2,-1,3,4,-2,5,6,7,-3,8,9,-4,10,11,12,-5,13,14,-6,15,16,-7,17,18,19,-8,20,21,-9,22,
%W A130527 23,24,-10,25,26,-11,27,28,-12,29,30,31,-13,32,33,-14,34,35,36,-15,37,38,-16,39,40,41,
%X A130527 -17,42,43,-18,44,45,-19,46,47,48
%N A130527 A permutation of the integers induced by the Beatty sequence for sqrt(2).
%C A130527 Another permutation of the integers is -A130527.
%F A130527 a(0)=0; a(n)=k if n=L(k); a(n)=-k if n=U(k), where L(k) = A001951(k) = Floor(k*sqrt(2)), U(k) = A001952(k) = 2k+L(k).
%e A130527 5=L(4), so a(5)=4.
%e A130527 6=U(2), so a(6)=-2.
%Y A130527 Cf. A001951, A001952, A130526.
%Y A130527 Adjacent sequences: A130524 A130525 A130526 this_sequence A130528 A130529 A130530
%Y A130527 Sequence in context: A125160 A009947 A026249 this_sequence A026366 A122164 A076632
%K A130527 sign
%O A130527 0,3
%A A130527 Clark Kimberling (ck6(AT)evansville.edu), Jun 02 2007

%I A026366
%S A026366 1,2,1,3,4,2,5,6,7,8,3,9,10,4,11,12,13,14,5,15,16,6,17,18,7,19,
%T A026366 20,8,21,22,23,24,9,25,26,10,27,28,29,30,11,31,32,12,33,34,13,
%U A026366 35,36,14,37,38,39,40,15,41,42,16,43,44,45,46,17,47,48,18,49
%N A026366 a(n) = a(m) if a(m) has already occurred exactly once and n = a(m)+2m, else a(n) = least positive integer that has not yet occurred.
%H A026366 A. S. Fraenkel, <a href="http://arXiv.org/abs/math.CO/9809074">Heap games, numeration systems and sequences</a>
%Y A026366 Cf. A045671.
%Y A026366 Adjacent sequences: A026363 A026364 A026365 this_sequence A026367 A026368 A026369
%Y A026366 Sequence in context: A009947 A026249 A130527 this_sequence A122164 A076632 A105646
%K A026366 nonn
%O A026366 1,2
%A A026366 Clark Kimberling (ck6(AT)evansville.edu)
%E A026366 Description corrected by Aviezri Fraenkel (fraenkel(AT)cs.curtin.edu.au)

%I A122164
%S A122164 0,1,1,2,1,3,4,2,6,3,9,12,6,18,9,27,36,18,54,27,81,108,54,162,81,243,
%T A122164 324,162,486,243,729,972,486,1458,729,2187,2916,1458,4374,2187,6561,
%U A122164 8748,4374,13122,6561,19683,26244,13122,39366,19683,59049,78732,39366
%N A122164 a(0) = 0, a(1) = 1, s = 0; for n >= 2, if a(n-1) is even and s = 0 then set a(n) = a(n-1)/2 and s = 1; otherwise set a(n) = a(n-1) + a(n-2) and s = 0.
%F A122164 For i >= 1, a(5i) = 3^i, a(5i+1) = 4*3^(i-1), a(5i+2) = 2*3^(i-1), a(5i+3) = 2*3^i, a(5i+4) = 3^i. - njas, Aug 06 2008
%F A122164 O.g.f.: x(-1-x-2x^2-x^3-3x^4-x^5+x^6)/(3x^5-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 07 2008]
%Y A122164 Cf. A000045, A122597. Records give A000792.
%Y A122164 Adjacent sequences: A122161 A122162 A122163 this_sequence A122165 A122166 A122167
%Y A122164 Sequence in context: A026249 A130527 A026366 this_sequence A076632 A105646 A059126
%K A122164 nonn
%O A122164 0,4
%A A122164 Philip van den Bossche (filip020667(AT)hotmail.com), Aug 03 2008
%E A122164 Edited by njas, Aug 06 2008
%E A122164 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 07 2008

%I A076632
%S A076632 1,1,1,2,1,3,4,2,9,6,12,23,1,46,45,47,136,43,229,314,144,771,484,1058,
%T A076632 2025,91,4140,3959,4321,12238,3597,20879,28072,13686
%N A076632 Solve 2^n - 2 = 7(x^2 - x) + (y^2 - y) for (x,y) with x>0, y>0; sequence gives values of y.
%C A076632 Euler (unpublished) showed there is a unique positive solution (x,y) for every positive n.
%D A076632 Engel, Problem-Solving Strategies.
%F A076632 Note that the equation is equivalent to 2^(n+2) = (2y-1)^2 + 7 (2x-1)^2, so it is related to norms of elements of the ring of integers in the quadratic field Q(sqrt(-7)) and Euler's claim presumably follows from unique factorization in that field. From this we can get a formula for the x's and y's: Let a(n) and b(n) be the unique rational numbers such that a(n) + b(n) sqrt(-7) = ((1 + sqrt(-7))/2)^n. I.e. a(n) = (((1 + sqrt(-7))/2)^n + ((1 - sqrt(-7))/2)^n)/2. - Dean Hickerson (dean(AT)math.ucdavis.edu), Oct 19, 2002.
%F A076632 a(n) = (1/sqrt(7))*2^(n/2)*abs(sin(n*t))+1/2, where t=arctan(sqrt(7)). - Paul Boddington (psb(AT)maths.warwick.ac.uk), Jan 23 2004
%o A076632 (PARI) p(n,x,y)=2^n-2-7*(x^2-x)-(y^2-y) a(n)=if(n<0,0,x=1; while(frac(real(component(polroots(p(n,x,y)),2)))>0,x++); x)
%Y A076632 Cf. A076631.
%Y A076632 Adjacent sequences: A076629 A076630 A076631 this_sequence A076633 A076634 A076635
%Y A076632 Sequence in context: A130527 A026366 A122164 this_sequence A105646 A059126 A059128
%K A076632 nonn,easy
%O A076632 1,4
%A A076632 Ed Pegg Jr. (ed(AT)mathpuzzle.com), Oct 17 2002
%E A076632 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 24 2002

%I A105646
%S A105646 1,2,1,3,4,3,1,2,1,4,3,4,2,1,2,4,3,4,1,2,1,3,4,3,1,2,1,2,1,2,4,3,4,2,1,
%T A105646 2,3,4,3,1,2,1,3,4,3,2,1,2,4,3,4,2,1,2,1,2,1,3,4,3,1,2,1,4,3,4,2,1,2,4,
%U A105646 3,4,1,2,1,3,4,3,1,2,1,3,4,3,1,2,1,3,4,3,2,1,2,4,3,4,2,1,2,3,4,3,1,2,1
%N A105646 Fixed point of the morphism 1 -> 121, 2 -> 343, 3 -> 434, 4 -> 212, starting from a(0) = 1.
%C A105646 Rectangular space-fill from Peano space-fill by row permutation of the digraph matrix: Characteristic polynomial: x^4-3*x^3-3*x+9.
%D A105646 F. M. Deking, "Recurrent Sets", Advances in Mathematics, vol. 44, no. 1, 1982, page 85, section 4.1
%F A105646 1->{1, 2, 1}, 2->{3, 4, 3}, 3->{4, 3, 4}, 4->{2, 1, 2}
%t A105646 Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {3, 4, 3}, 3 -> {4, 3, 4}, 4 -> {2, 1, 2}}] &, {1}, 4]] (* Robert G. Wilson v *)
%Y A105646 Adjacent sequences: A105643 A105644 A105645 this_sequence A105647 A105648 A105649
%Y A105646 Sequence in context: A026366 A122164 A076632 this_sequence A059126 A059128 A050273
%K A105646 nonn
%O A105646 0,2
%A A105646 Roger Bagula (rlbagulatftn(AT)yahoo.com), May 03 2005
%E A105646 Edited by Robert G. Wilson v (rgwv(at)rgwv.com), Jan 24 2006

%I A059126
%S A059126 1,2,1,3,4,3,1,2,1,5,6,5,1,2,1,3,4,3,1,2,1,7,8,7,1,2,1,3,4,3,1,2,1,5,6,
%T A059126 5,1,2,1,3,4,3,1,2,1,9,10,9,1,2,1,3,4,3,1,2,1,5,6,5,1,2,1,3,4,3,1,2,1,
%U A059126 7,8,7,1,2,1,3,4,3,1,2,1,5,6,5,1,2,1,3,4,3,1,2,1,11,12,11,1,2,1,3,4,3
%N A059126 A hierarchical sequence (W2{2} according to the description in the attached file - see link).
%H A059126 J. Wallgren, <a href="http://www.research.att.com/~njas/sequences/A059126.txt">Hierarchical sequences</a>
%Y A059126 Adjacent sequences: A059123 A059124 A059125 this_sequence A059127 A059128 A059129
%Y A059126 Sequence in context: A122164 A076632 A105646 this_sequence A059128 A050273 A122530
%K A059126 easy,nonn
%O A059126 0,2
%A A059126 Jonas Wallgren (jonwa(AT)ida.liu.se), Jan 19 2001

%I A059128
%S A059128 1,2,1,3,4,3,1,2,1,5,6,5,7,8,7,5,6,5,1,2,1,3,4,3,1,2,1,9,10,9,11,12,11,
%T A059128 9,10,9,1,2,1,3,4,3,1,2,1,5,6,5,7,8,7,5,6,5,1,2,1,3,4,3,1,2,1,13,14,13,
%U A059128 15,16,15,13,14,13,1,2,1,3,4,3,1,2,1,5,6,5,7,8,7,5,6,5,1,2,1,3,4,3,1,2
%N A059128 A hierarchical sequence (W3{2,2} - see A059126).
%H A059128 J. Wallgren, <a href="http://www.research.att.com/~njas/sequences/A059126.txt">Hierarchical sequences</a>
%Y A059128 Adjacent sequences: A059125 A059126 A059127 this_sequence A059129 A059130 A059131
%Y A059128 Sequence in context: A076632 A105646 A059126 this_sequence A050273 A122530 A022466
%K A059128 easy,nonn
%O A059128 0,2
%A A059128 Jonas Wallgren (jonwa(AT)ida.liu.se), Jan 19 2001

%I A050273
%S A050273 1,2,1,3,4,3,1,2,5,6,4,7,3,1,8,5,2,3,9,4,10,8,1,6,11,5,12,8,3,4,13,7,2,
%T A050273 6,14,1,15,3,8,16,7,12,4,5,17,9,18,15,8,1,12,19,2,10,20,3,9,5,6,21,4,8,
%U A050273 22,11,16,3,10,23,1,24,12,6,7,25,8,11,16,26,4,13,2,15,27,12,5,28
%N A050273 Smallest value a for Diophantine 1-triples (a,b,c) ordered by smallest c,b.
%H A050273 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantusProperty.html">Link to a section of The World of Mathematics.</a>
%Y A050273 Cf. A050274, A050275.
%Y A050273 Adjacent sequences: A050270 A050271 A050272 this_sequence A050274 A050275 A050276
%Y A050273 Sequence in context: A105646 A059126 A059128 this_sequence A122530 A022466 A133310
%K A050273 nonn
%O A050273 1,2
%A A050273 Eric Weisstein (eric(AT)weisstein.com)

%I A122530
%S A122530 1,2,1,3,4,3,2,5,4,3,2,6,5,4,7,6,8,7,6,5,4,9,8,7,6,10,9,8,11,10,12,11,
%T A122530 10,9,8,7,13,12,11,10,9,14,13,12,11,15,14,13,12,11,10,9,16,15,14,13,12,
%U A122530 11,17,16,15,14,13,12,11,10,18,17,16,15,14,13,12,19,18,17,16,15,14,20
%N A122530 Decreasing runs. The size of each run is given by the sequence itself. First run starts with "1", second with "2", third with "3", fourth with "4", etc.
%C A122530 Terms computed by Gilles Sadowski
%e A122530 S is the sequence and R is the size of each run:
%e A122530 S=1,2,1,3,4,3,2,5,4,3,2,6,5,4,7,6,8,7,6,5,4,9,8,7,6,10,9,8,11,10...
%e A122530 S=1,(2,1),(3),(4,3,2),(5,4,3,2),(6,5,4),(7,6),(8,7,6,5,4)...
%e A122530 R=1 2 1 3 4 3 2 5... = sequence S
%Y A122530 Adjacent sequences: A122527 A122528 A122529 this_sequence A122531 A122532 A122533
%Y A122530 Sequence in context: A059126 A059128 A050273 this_sequence A022466 A133310 A077608
%K A122530 base,easy,nonn
%O A122530 1,2
%A A122530 Eric Angelini (eric.angelini(AT)kntv.be), Sep 17 2006

%I A022466
%S A022466 1,2,1,3,4,3,4,6,8,12,13,18,24,33,39,52,67,88,113,155,211,264,331,
%T A022466 455,596,762,1000,1288
%N A022466 Number of 1's in n-th term of A007651.
%Y A022466 Adjacent sequences: A022463 A022464 A022465 this_sequence A022467 A022468 A022469
%Y A022466 Sequence in context: A059128 A050273 A122530 this_sequence A133310 A077608 A002124
%K A022466 nonn
%O A022466 1,2
%A A022466 Clark Kimberling (ck6(AT)evansville.edu)

%I A133310
%S A133310 1,2,1,3,4,3,5,6,5,7,8,7,9,10,9,11,12,11,13,14,13,15,16,15,17,18,17,19,
%T A133310 20,19,21,22,21,23,24,23,25,26,25,27,28,27,29,30,29,31,32,31,33,34,33,
%U A133310 35,36,35,37,38,37,39,40,39,41,42,41,43,44,43,45,46,45,47,48,47,49,50
%N A133310 a(3n) = 2n+1, a(3n+1) = 2n+2, a(3n+2) = 2n+1.
%Y A133310 Cf. A130823(1, 1, 1, 3, 3, 3).
%Y A133310 Adjacent sequences: A133307 A133308 A133309 this_sequence A133311 A133312 A133313
%Y A133310 Sequence in context: A050273 A122530 A022466 this_sequence A077608 A002124 A097564
%K A133310 nonn
%O A133310 0,2
%A A133310 Paul Curtz (bpcrtz(AT)free.fr), Oct 18 2007

%I A077608
%S A077608 1,0,0,1,0,1,1,1,2,1,3,4,3,7,7,8,14,15,21,28,33,47,58,75,103,125,167,
%T A077608 220,275,370,474,610,806,1028,1347,1752,2253,2954,3812,4944,6451,8329,
%U A077608 10841,14077,18226,23720,30745,39903,51857,67214,87313
%N A077608 Number of compositions of n into twin primes (i.e. primes that are members of a twin prime pair, like 3,5,7,11,13, but not 2 or 23).
%H A077608 P. Flajolet, <a href="http://algo.inria.fr/flajolet/">Publications</a>
%F A077608 A77608 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j)* subs([true=1, false=0], evalb(isprime(ithprime(j)-2) or isprime(ithprime(j)+2))), j=2..n+2)), z=0, n+1), z, n): end;
%e A077608 a(15)=8 since 15=11+7=7+11=5+13=13+5=3+5+7=3+7+5=5+3+7=5+7+3=7+3+5=7+5+3, and 3,5,7,11 belong to twin pairs.
%p A077608 A077608 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j)* subs([true=1,false=0],evalb(isprime(ithprime(j)-2) or isprime(ithprime(j)+2))),j=2..n+2)),z=0,n+1),z,n): end;
%Y A077608 Cf. A002124, A023360.
%Y A077608 Adjacent sequences: A077605 A077606 A077607 this_sequence A077609 A077610 A077611
%Y A077608 Sequence in context: A122530 A022466 A133310 this_sequence A002124 A097564 A128270
%K A077608 nonn
%O A077608 0,9
%A A077608 Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Nov 11 2002

%I A002124 M0154 N0062
%S A002124 1,0,0,1,0,1,1,1,2,1,3,4,3,7,7,8,14,15,21,28,33,47,58,76,103,125,169,220,277,373,
%T A002124 476,616,810,1037,1361,1763,2279,2984,3846,5006,6521,8428,10983,14249,
%U A002124 18480,24048,31178,40520,52635,68281,88765,115211,149593,194381,252280,327696,425587,552527,717721
%N A002124 Number of compositions of n into a sum of odd primes.
%C A002124 Arises in studying the Goldbach conjecture.
%C A002124 The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by S. Plouffe in his 1992 dissertation is wrong.
%D A002124 P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [The sequence i_n]
%H A002124 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b002124.txt">Table of n, a(n) for n = 0..1000</a>
%H A002124 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002124 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A002124 a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]
%p A002124 A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j),j=2..n)),z=0,n+1),z,n) end;
%p A002124 M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n],n=0..M)]; [njas, after MacMahon, Dec 03 2006] [Used in A002125]
%Y A002124 Cf. A002125, A023360, A024939, A077608.
%Y A002124 Adjacent sequences: A002121 A002122 A002123 this_sequence A002125 A002126 A002127
%Y A002124 Sequence in context: A022466 A133310 A077608 this_sequence A097564 A128270 A097003
%K A002124 nonn
%O A002124 0,9
%A A002124 njas
%E A002124 Better description and more terms from Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Nov 11 2002
%E A002124 Edited by njas, Dec 03 2006

%I A097564
%S A097564 0,1,1,2,1,3,4,3,7,10,7,17,24,17,41,58,41,99,140,99,239,338,239,577,816,
%T A097564 577,1393,1970,1393,3363,4756,3363,8119,11482,8119,19601,27720,19601,
%U A097564 47321,66922,47321,114243,161564,114243,275807,390050,275807,665857
%N A097564 a(0) = 0, a(1) = 1, a(n) = Mod[a(n-1),2]*a(n-1) + a(n-2) for n > 1.
%C A097564 The sequences a(2), a(5), ... a(1+3*n) ... and a(4), a(7), ... a(4 + 3n) ... are both A001333 (numerators of continued fraction convergents to sqrt(2)). The sequence a(0), a(3), a(6), ... a(3+3*n) ... is 2 times A000129 (the Pell nos. or the denominators of continued fraction convergents to sqrt(2)., also is A052542 starting w/ offset 1.
%Y A097564 Adjacent sequences: A097561 A097562 A097563 this_sequence A097565 A097566 A097567
%Y A097564 Sequence in context: A133310 A077608 A002124 this_sequence A128270 A097003 A109447
%K A097564 nonn
%O A097564 0,4
%A A097564 Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 27 2004

%I A128270
%S A128270 2,1,3,4,3,12,3,48,3,320,3,512,135,256,243,5120,243,8192,27,5120,27,
%T A128270 2048,135,5120,1701,8192,2187,4096,2187,1024,6561,1792,1215,25088,243,
%U A128270 62720,27,313600,27,1568000,243,2508800,243,6272000,243,31360000,27
%V A128270 2,1,-3,4,-3,12,-3,48,-3,320,-3,512,-135,256,-243,5120,-243,8192,-27,
%W A128270 5120,-27,2048,-135,5120,-1701,8192,-2187,4096,-2187,1024,-6561,1792,
%X A128270 -1215,25088,-243,62720,-27,313600,-27,1568000,-243,2508800,-243,6272000
%N A128270 a(n) = the numerator of b(n): {b(n)} is such that the continued fraction (of rational terms) [b(1);b(2),...,b(n)] equals the n-th prime, for every positive integer n.
%H A128270 Diana Mecum, <a href="http://www.research.att.com/~njas/sequences/b128270.txt">Table of n, a(n) for n = 1..500</a>
%e A128270 b(n): 2, 1, -3/2, 4, -3/4, 12, -3/16,...
%e A128270 The 4th prime, 7, equals [b(1);b(2),b(3),b(4)] = 2 +1/(1 +1/(-3/2 +1/4)).
%e A128270 The 5th prime, 11, equals [b(1);b(2),b(3),b(4),b(5)] = 2 +1/(1 +1/(-3/2 +1/(4 -4/3))).
%Y A128270 Cf. A128271.
%Y A128270 Adjacent sequences: A128267 A128268 A128269 this_sequence A128271 A128272 A128273
%Y A128270 Sequence in context: A077608 A002124 A097564 this_sequence A097003 A109447 A088261
%K A128270 frac,sign
%O A128270 1,1
%A A128270 Leroy Quet (qq-quet(AT)mindspring.com), Feb 22 2007
%E A128270 More terms from Diana Mecum (diana.mecum(AT)gmail.com), Jun 24 2007

%I A097003
%S A097003 1,1,2,1,3,4,4,1,3,4,10,3,3,11,16,1,7,10,13,25,10,5,79,58,99,100,94,92,
%T A097003 59,37,54,1
%N A097003 Function A062402[x]=phi[sigma[x]] is iterated. a(n) is the number of distinct terms arising in the trajectory of 2^n; a(n)=t(n)+c(n)=t+c, where t is the number of transient terms, c is the number of recurrent terms [in the terminal cycle].
%C A097003 Concerning this sequence and A097004, A096994, A096995: in all 4 cases the initial value is 2^n and a certain function is iterated. They differ either in the function or in what is computed for that iteration.
%C A097003 Glossary: t+c = total count of transient+cycle terms, t = count of transient terms
%C A097003 Sequence 1: A062401 is iterated t+c is computed => this sequence
%C A097003 Sequence 2: A062402 is iterated t+c is computed => A097004
%C A097003 Sequence 3: A062401 is iterated t is computed => A096994
%C A097003 Sequence 4: A062402 is iterated t is computed => A096995
%e A097003 n=13: 2^n=8192, trajectory ={8192, 10584, 8640, 8064, 6144, [3456, 2560, 1800, 2880, 3024, 3840], 3456, 2560, ..}, t+c=a(13)=5+6=11;
%t A097003 EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 20]]], {w, 1, 256}]
%Y A097003 Cf. A000010, A000203, A062402, A096852.
%Y A097003 Adjacent sequences: A097000 A097001 A097002 this_sequence A097004 A097005 A097006
%Y A097003 Sequence in context: A002124 A097564 A128270 this_sequence A109447 A088261 A033779
%K A097003 nonn
%O A097003 0,3
%A A097003 Labos E. (labos(AT)ana.sote.hu), Jul 21 2004

%I A109447
%S A109447 1,2,1,3,4,4,1,10,5,6,20,6,1,21,35,7,8,56,56,8,1,36,126,84,9,10,120,252,
%T A109447 120,10,1,55,330,462,165,11,12,220,792,792,220,12,1,78,715,1716,1287,
%U A109447 286,13,14,364,2002,3432,2002,364,14,1,105,1365,5005,6435,3003,455,15
%N A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.
%e A109447 Starred terms in Pascal's triangle (A007318), read by rows:
%e A109447 1;
%e A109447 1*, 1;
%e A109447 1, 2*, 1;
%e A109447 1*, 3, 3*, 1;
%e A109447 1, 4*, 6, 4*, 1;
%e A109447 1*, 5, 10*, 10, 5*, 1;
%e A109447 1, 6*, 15, 20*, 15, 6*, 1;
%e A109447 1*, 7, 21*, 35, 35*, 21, 7*, 1;
%e A109447 1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
%e A109447 1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
%e A109447 1; 2; 1, 3; 4, 4; 1, 10, 5; 6, 20, 6; 1, 21, 35, 7; 8, 56, 56, 8; ....
%t A109447 Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)
%Y A109447 Cf. A109446.
%Y A109447 Adjacent sequences: A109444 A109445 A109446 this_sequence A109448 A109449 A109450
%Y A109447 Sequence in context: A097564 A128270 A097003 this_sequence A088261 A033779 A033799
%K A109447 easy,nonn,tabf
%O A109447 0,2
%A A109447 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2005
%E A109447 More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Aug 30 2005

%I A088261
%S A088261 1,1,1,1,2,1,3,4,4,2,1,2,1,2,1,1,2,4,3,1,1,3,1,3,4,1,1,2,2,1,2,1,1,2,4,
%T A088261 3,1,1,1,1,5,1,5,1,2,2,1,3,3,1,4,1,2,4,2,4,2,2,5,11,1,1,4,4,2,1,3,3,1,6,
%U A088261 4,2,1,13,1,5,3,1,5,6,1,2,2,3,4,1,1,4,2,1,5,1,1,5,1,2,4,2,6,1,2,3,3,1,4
%N A088261 Smallest number prefixed with corresponding term of the sequence of numbers == 1, 3, 7 or 9 (mod 10), i.e. (1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, ...) that yields a prime.
%e A088261 The term corresponding to 61 is 4 as 161, 261 and 361 are composite.
%Y A088261 Adjacent sequences: A088258 A088259 A088260 this_sequence A088262 A088263 A088264
%Y A088261 Sequence in context: A128270 A097003 A109447 this_sequence A033779 A033799 A085985
%K A088261 base,easy,nonn
%O A088261 0,5
%A A088261 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 28 2003
%E A088261 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jul 27 2005

%I A033779
%S A033779 1,1,1,2,1,3,4,4,3,3,6,5,9,6,5,11,10,13,11,9,15,17,18,12,
%T A033779 15,22,19,29,18,18,33,30,34,27,28,41,40,45,33,39,55,44,
%U A033779 64,39,42,71,55,66,54,55,76
%N A033779 Product t2(q^d); d | 20, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033779 Adjacent sequences: A033776 A033777 A033778 this_sequence A033780 A033781 A033782
%Y A033779 Sequence in context: A097003 A109447 A088261 this_sequence A033799 A085985 A088267
%K A033779 nonn
%O A033779 0,4
%A A033779 njas

%I A033799
%S A033799 1,1,1,2,1,3,4,4,4,4,7,7,10,9,9,15,13,16,17,14,24,23,23,
%T A033799 23,26,36,31,40,34,38,55,46,52,52,53,75,67,69,71,80,100,
%U A033799 85,104,91,98,136,110,121,129,126,167
%N A033799 Product t2(q^d); d | 40, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033799 Adjacent sequences: A033796 A033797 A033798 this_sequence A033800 A033801 A033802
%Y A033799 Sequence in context: A109447 A088261 A033779 this_sequence A085985 A088267 A117407
%K A033799 nonn
%O A033799 0,4
%A A033799 njas

%I A085985
%S A085985 0,1,2,1,3,4,5,1,2,6,7,8,9,10,11,1,12,13,14,15,16,17,18,19,3,20,2,21,22,
%T A085985 23,24,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,4,41,42,43,44,
%U A085985 45,46,47,48,49,50,51,52,53,54,1,55,56,57,58,59,60,61,62,63,64
%N A085985 a(n) = A049084(A085818(n)).
%C A085985 a(A085971(n))=A000027(n) and for all k>1: a(A000040(n)^k)=A000027(n).
%Y A085985 Adjacent sequences: A085982 A085983 A085984 this_sequence A085986 A085987 A085988
%Y A085985 Sequence in context: A088261 A033779 A033799 this_sequence A088267 A117407 A092790
%K A085985 nonn
%O A085985 1,3
%A A085985 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 06 2003

%I A088267
%S A088267 2,1,3,4,5,1,5,1,2,1,6,9,18,9,5,4,2,7,3,1,6,3,11,9,20,6,32,16,11,13,11,
%T A088267 1,6,15,6,21,18,1,5,13,14,1,6,18,15,9,12,3,50,19,56,12,9,3,5,66,27,1,8,
%U A088267 24,11,9,11,13,24,79,3,43,26,19,11,4,39,3,45,9,11,6,6,15,2,1,41,18,5,19
%N A088267 Smallest number that on prefixing with the n-th term of A088265 yields prime.
%e A088267 a(13) = 18 as A088265(13) = 10001 and 1810001 is a prime while 110001, 210001 to 1710001 are all composite.
%Y A088267 Cf. A088265, A088267.
%Y A088267 Adjacent sequences: A088264 A088265 A088266 this_sequence A088268 A088269 A088270
%Y A088267 Sequence in context: A033779 A033799 A085985 this_sequence A117407 A092790 A082470
%K A088267 base,nonn
%O A088267 0,1
%A A088267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 28 2003
%E A088267 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jul 28 2005

%I A117407
%S A117407 1,2,1,3,4,5,2,6,7,3,8,9,10,4,11,12,13,5,14,15,6,16,17,18,7,19,20,8,21,
%T A117407 22,23,9,24,25,26,10,27,28,11,29,30,31,12,32,33,34,13,35,36,14,37,38,39,
%U A117407 15,40,41,16,42,43,44,17,45,46,47,18,48,49,19,50,51,52,20,53,54,21,55
%N A117407 a(n) = j if n is T(j), else a(n) = k if n is U(k), where T is a Beatty sequence based on (sqrt(5)+5)/2 (A054770) and U is its complement (A063732).
%C A117407 Every positive integer occurs exactly twice. Taking a Lucas number (A000032) of terms L(n) starting at a(0), the last two terms are a pair of Fibonacci numbers (A000045). If n is even, then the last two terms are F(n+1) followed by F(n-1), if n is odd they are F(n-1) followed by F(n+1), where F is the Fibonacci sequence. For example, the first L(4) = 7 terms of this sequence are (1,2,1,3,4,5,2), and the last members are 5 and 2 which are equal to F(5) and F(3). Note also that L(n) = F(n-1) + F(n+1).
%e A117407 a(9) = 3 because 9 = T(3).
%Y A117407 Cf. A026272, A026242.
%Y A117407 Adjacent sequences: A117404 A117405 A117406 this_sequence A117408 A117409 A117410
%Y A117407 Sequence in context: A033799 A085985 A088267 this_sequence A092790 A082470 A101204
%K A117407 nonn
%O A117407 0,2
%A A117407 Casey Mongoven (cm(AT)caseymongoven.com), Mar 13 2006

%I A092790
%S A092790 2,1,3,4,5,3,6,7,6,6,9,11,9,5,10,9,10,9,9,8,9,9,11,8,10,10,12,16,12,10
%N A092790 Duplicate of A082470.
%Y A092790 Adjacent sequences: A092787 A092788 A092789 this_sequence A092791 A092792 A092793
%Y A092790 Sequence in context: A085985 A088267 A117407 this_sequence A082470 A101204 A035043
%K A092790 dead
%O A092790 1,1

%I A082470
%S A082470 2,1,3,4,5,3,6,7,6,6,9,11,9,5,10,9,10,9,9,8,9,9,11,8,10,10,12,16,12,10,10,
%T A082470 13,14,14,16,11,12,9,15,10,9,8,12,9,10,6,8,7,14,13,10,21,15,9,13,11,9,
%U A082470 19,12,13,16,11,19,17,9,13
%N A082470 Number of k >= 0 such that k! + prime(n) is prime.
%C A082470 k! + p is composite for k >= p since p divides k! for k >= p.
%e A082470 For n = 4, 3!+7 = 13, 4!+7=31, 5!+7=127 and 6!+7 = 727 are the 4 primes in n!+7
%p A082470 for i from 2 to 50 do ctr := 0: for j from 2 to ithprime(i)-1 do if isprime(j!+ithprime(i))=true then ctr := ctr+1 fi od; print(ctr); od;
%o A082470 (PARI) nfactppct(n) = { forprime(p=1,n, c=0; for(x=0,n,y=x!+p;if(isprime(y),c++) ); print1(c",") ) } - Cino Hilliard (hillcino368(AT)gmail.com), Apr 15 2004
%Y A082470 Adjacent sequences: A082467 A082468 A082469 this_sequence A082471 A082472 A082473
%Y A082470 Sequence in context: A088267 A117407 A092790 this_sequence A101204 A035043 A058684
%K A082470 nonn,more
%O A082470 2,1
%A A082470 Jeff Burch (gburch(AT)erols.com), Apr 27 2003
%E A082470 Edited by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 01 2006

%I A101204
%S A101204 1,0,1,1,0,1,1,1,2,1,3,4,5,4,1,9,16,22,16,7,1,32,75,112,86,41,10,1,133
%N A101204 Triangle read by rows: T(n,k) = number of planar trivalent (or cubic) multigraphs with 2n nodes and exactly k double bonds, for 0 <= k <= n.
%C A101204 The entries in the first two rows are "by convention".
%D A101204 A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
%e A101204 Triangle begins
%e A101204 1
%e A101204 0 1
%e A101204 1 0 1
%e A101204 1 1 2 1
%e A101204 3 4 5 4 1
%e A101204 9 16 22 16 7 1
%e A101204 32 75 112 86 41 10 1
%e A101204 133 ...
%Y A101204 Row sums give A005966. First column is A005964 (trivalent connected planar graphs with 2n nodes). Second and third columns give A101205, A101206.
%Y A101204 Adjacent sequences: A101201 A101202 A101203 this_sequence A101205 A101206 A101207
%Y A101204 Sequence in context: A117407 A092790 A082470 this_sequence A035043 A058684 A109920
%K A101204 nonn,tabl
%O A101204 0,9
%A A101204 njas, Dec 13 2004

%I A035043
%S A035043 0,2,1,3,4,5,6,7,8,9,20,22,21,23,24,25,26,27,28,29,10,12,11,13,14,
%T A035043 15,16,17,18,19,30,32,31,33,34,35,36,37,38,39,40,42,41,43,44,45,46,
%U A035043 47,48,49,50,52,51,53,54,55,56,57,58,59,60,62,61,63,64,65,66,67,68
%N A035043 Exchange 1 and 2!.
%Y A035043 Adjacent sequences: A035040 A035041 A035042 this_sequence A035044 A035045 A035046
%Y A035043 Sequence in context: A092790 A082470 A101204 this_sequence A058684 A109920 A109919
%K A035043 nonn,base,easy
%O A035043 0,2
%A A035043 njas

%I A058684
%S A058684 1,0,2,1,3,4,5,6,7,11,15,17,22,24,34,40,48,56,69,84,104,118,144,164,
%T A058684 200
%N A058684 McKay-Thompson series of class 45A for Monster.
%D A058684 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058684 T45A = 1/q + 2*q + q^2 + 3*q^3 + 4*q^4 + 5*q^5 + 6*q^6 + 7*q^7 + 11*q^8 + ...
%Y A058684 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058684 Adjacent sequences: A058681 A058682 A058683 this_sequence A058685 A058686 A058687
%Y A058684 Sequence in context: A082470 A101204 A035043 this_sequence A109920 A109919 A082750
%K A058684 nonn
%O A058684 -1,3
%A A058684 njas, Nov 27, 2000

%I A109920
%S A109920 1,2,1,3,4,5,6,7,360,11,12,13,1680,17,18,19,4620,23,491400,29,30,31,
%T A109920 1884960,37,29640,41,42,43,45540,47,12994800,53,45821160,59,60,61,
%U A109920 89369280,67,164220,71,72,73,211110900,79,265680,83,195878760,89
%N A109920 a(1) = 1, then LCM of consecutive composite numbers sandwitched between primes.
%F A109920 a(2n) = prime(n) a(2n+1)= LCM of composite numbers between prime(n) and prime(n+1). a(1) = a(3) = 1 by choice.
%p A109920 A109920 := proc(n) local p; if n mod 2 = 0 then ithprime(n/2) ; elif n = 1 then 1 ; else p := ithprime((n-1)/2) ; lcm(seq(i,i=p+1..nextprime(p)-1)) ; fi ; end: for n from 1 to 80 do printf("%d, ",A109920(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2007
%Y A109920 Cf. A109919.
%Y A109920 Adjacent sequences: A109917 A109918 A109919 this_sequence A109921 A109922 A109923
%Y A109920 Sequence in context: A101204 A035043 A058684 this_sequence A109919 A082750 A048212
%K A109920 easy,nonn
%O A109920 1,2
%A A109920 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 16 2005
%E A109920 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2007

%I A109919
%S A109919 1,2,1,3,4,5,6,7,720,11,12,13,3360,17,18,19,9240,23,11793600,29,30,31,
%T A109919 45239040,37,59280,41,42,43,91080,47,311875200,53,549853920,59,60,61,
%U A109919 1072431360,67,328440,71,72,73,2533330800,79,531360,83,4701090240,89
%N A109919 a(1) = 1, then product of consecutive composite numbers sandwitched between primes.
%C A109919 a(1) = a(3) = 1 as empty product is defined to be 1.
%F A109919 a(2n) = prime(n) a(2n+1)= product of composite numbers between prime(n) and prime(n+1).
%F A109919 a(2n) = A000040(n). a(2n+1) = A072472(n)/A000040(n+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2007
%p A109919 A109919 := proc(n) local p; if n mod 2 = 0 then ithprime(n/2) ; elif n = 1 then 1 ; else p := ithprime((n-1)/2) ; mul(i,i=p+1..nextprime(p)-1) ; fi ; end: for n from 1 to 80 do printf("%d, ",A109919(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2007
%Y A109919 Cf. A109920.
%Y A109919 Cf. A072472.
%Y A109919 Adjacent sequences: A109916 A109917 A109918 this_sequence A109920 A109921 A109922
%Y A109919 Sequence in context: A035043 A058684 A109920 this_sequence A082750 A048212 A077159
%K A109919 easy,nonn
%O A109919 1,2
%A A109919 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 16 2005
%E A109919 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2007

%I A082750
%S A082750 2,1,3,4,5,7,6,8,9,17,10,11,12,13,43,14,15,16,18,19,21,20,22,23,24,25,
%T A082750 26,39,27,28,29,30,31,32,33,47,34,35,36,37,38,40,41,42,59,44,45,46,48,
%U A082750 49,50,51,52,53,71,54,55,56,57,58,60,61,62,63,64,99,65,66,67,68,69,70
%N A082750 Triangle read by rows, in which the n-th row contains n smallest numbers not occurring earlier such that the concatenation of terms of a row yields a prime.
%e A082750 2
%e A082750 1 3
%e A082750 4 5 7
%e A082750 6 8 9 17
%e A082750 10 11 12 13 43
%e A082750 ...
%Y A082750 Cf. A082751.
%Y A082750 Adjacent sequences: A082747 A082748 A082749 this_sequence A082751 A082752 A082753
%Y A082750 Sequence in context: A058684 A109920 A109919 this_sequence A048212 A077159 A075379
%K A082750 base,nonn
%O A082750 1,1
%A A082750 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 17 2003
%E A082750 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 23 2003

%I A048212
%S A048212 2,1,3,4,5,7,6,10,11,13,8,14,18,19,21,9,17,23,27,28,30,12,21,29,35,39,
%T A048212 40,42,15,27,36,44,50,54,55,57,16,31,43,52,60,66,70,71,73,20,36,51,63,
%U A048212 72,80,86,90,91,93,22,42,58,73,85,94,102,108
%N A048212 Triangular array T read by rows: T(i,j)=b(i+1)-b(i+1-j); j=1,2,...,i; i=1,2,3,...; b=A004979.
%e A048212 Rows: {2}; {1,3}; {4,5,7}; ...
%Y A048212 Adjacent sequences: A048209 A048210 A048211 this_sequence A048213 A048214 A048215
%Y A048212 Sequence in context: A109920 A109919 A082750 this_sequence A077159 A075379 A122514
%K A048212 nonn,tabl
%O A048212 1,1
%A A048212 Clark Kimberling (ck6(AT)evansville.edu)

%I A077159
%S A077159 2,1,3,4,5,7,9,11,13,15,6,8,12,14,16,17,19,23,25,29,31,10,18,20,22,24,
%T A077159 26,27,21,33,35,37,39,41,43,45,28,32,34,38,40,44,46,47,49,51,53,57,59,
%U A077159 61,63,67,69,71,73,30,36,42,48,50,52,54,56,58,60,62,55,65,77,79,83,85
%N A077159 Triangle in which n-th row contains smallest n numbers coprime to n which have not occurred earlier.
%C A077159 A permutation of the natural numbers.
%e A077159 Triangle begins:
%e A077159 2
%e A077159 1 3
%e A077159 4 5 7
%e A077159 9 11 13 15
%e A077159 6 8 12 14 16
%e A077159 17 19 23 25 29 31
%e A077159 10 18 20 22 23 24 26
%e A077159 ...
%Y A077159 Cf. A077160, A077161, A077162.
%Y A077159 Adjacent sequences: A077156 A077157 A077158 this_sequence A077160 A077161 A077162
%Y A077159 Sequence in context: A109919 A082750 A048212 this_sequence A075379 A122514 A130077
%K A077159 easy,nonn,tabl
%O A077159 1,1
%A A077159 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 31 2002
%E A077159 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

%I A075379
%S A075379 2,1,3,4,5,8,6,9,7,12,10,16,11,18,13,20,14,24,15,25,17,27,19,28,21,32,
%T A075379 22,36,23,40
%N A075379 Erroneous version of A088610.
%Y A075379 Adjacent sequences: A075376 A075377 A075378 this_sequence A075380 A075381 A075382
%Y A075379 Sequence in context: A082750 A048212 A077159 this_sequence A122514 A130077 A080412
%K A075379 dead
%O A075379 1,1

%I A122514
%S A122514 0,1,0,2,1,3,4,5,10,11,21,27,43,64,92,144,205,316,462,693,1035,1532,
%T A122514 2301,3406,5099,7581,11303,16855,25088,37432,55728,83097,123800,184490,
%U A122514 274969,409683,610628,909845,1355970,2020635,3011157,4487395,6686979
%N A122514 Expansion of x/(1 - 2*x^2 - x^3 + x^4).
%Y A122514 Cf. A006054, A006053.
%Y A122514 Adjacent sequences: A122511 A122512 A122513 this_sequence A122515 A122516 A122517
%Y A122514 Sequence in context: A048212 A077159 A075379 this_sequence A130077 A080412 A098164
%K A122514 nonn
%O A122514 1,4
%A A122514 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2006

%I A130077
%S A130077 0,2,1,3,4,6,5,6,7,9,8,11,13,14,12,16,15,17,16,18,19,21,20,21,22,24,23,
%T A130077 27,27,30,27,29,31,33,32,34,35,37,36,37,38,40,39,42,44,45,43,50,46,48,
%U A130077 47,49,50,52,51,52,53,55,54,59,58,62,58,60,63,65,64,66,67,69,68,69,70
%N A130077 Largest x such that 2^x divides A001623(n), the number of reduced three-line Latin rectangles.
%D A130077 John Riordan, A recurrence relation for three-line Latin rectangles, Amer. Math. Monthly, 59 (1952), pp. 159-162.
%Y A130077 Cf. A001623, A130078, A130079.
%Y A130077 Adjacent sequences: A130074 A130075 A130076 this_sequence A130078 A130079 A130080
%Y A130077 Sequence in context: A077159 A075379 A122514 this_sequence A080412 A098164 A060214
%K A130077 nonn
%O A130077 3,2
%A A130077 Douglas Stones (dssto1(AT)student.monash.edu.au), May 06 2007

%I A080412
%S A080412 0,2,1,3,4,6,5,7,8,10,9,11,12,14,13,15,16,18,17,19,20,22,21,23,24,26,25,
%T A080412 27,28,30,29,31,32,34,33,35,36,38,37,39,40,42,41,43,44,46,45,47,48,50,
%U A080412 49,51,52,54,53,55,56,58,57,59,60,62,61,63,64,66,65,67,68,70,69,71,72
%N A080412 Exchange rightmost two binary digits of n>1; a(0)=0, a(1)=2.
%C A080412 For n>3: a(n) = 4*floor(n/4) + a(n mod 4); self-inverse permutation of natural numbers: a(a(n))=n.
%H A080412 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A080412 a(20)=a('101'00')='101'00'=20; a(21)=a('101'01')='101'10'=22.
%e A080412 a(2)=a('10')='01'=2; a(3)=a('11')='11'=3;
%Y A080412 Cf. A007088, A004442, A080413, A080414.
%Y A080412 Adjacent sequences: A080409 A080410 A080411 this_sequence A080413 A080414 A080415
%Y A080412 Sequence in context: A075379 A122514 A130077 this_sequence A098164 A060214 A030133
%K A080412 nonn,nice
%O A080412 0,2
%A A080412 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 17 2003

%I A098164
%S A098164 0,2,1,3,4,6,5,7,8,21,9,20,11,22,13,24,15,26,17,28,19,40,31,42,33,44,35,
%T A098164 46,37,48,39,60,51,62,53,64,55,66,57,68,59,80,71,82,73,84,75,86,77,88,
%U A098164 79,201,10,23,12,25,14,27,16,29,18,41,30,43,32,45,34,47
%N A098164 Smallest available number fitting the infinite repeating pattern of digits even/even/odd/odd/even/even/odd/odd/...
%Y A098164 Adjacent sequences: A098161 A098162 A098163 this_sequence A098165 A098166 A098167
%Y A098164 Sequence in context: A122514 A130077 A080412 this_sequence A060214 A030133 A139374
%K A098164 base,easy,nonn
%O A098164 0,2
%A A098164 Eric Angelini (eric.angelini(AT)kntv.be), Oct 25 2004

%I A060214
%S A060214 2,1,3,4,7,1,1,1,8,2,9,4,7,7,6,1,2,3,1,9,9,3,2,2,5,2,1,8,4,3,1,3,6,4,2,
%T A060214 2,0,7,3,5,7,1,5,7,7,8,9,3,4,9,1,5,1,2,7,2,4,4,7,6,3,9,6,0,3,6,4,0,7,9,
%U A060214 1,0,3,6,8,2,1,6,7,7,6,1,2,7,1,4,4,3,4,3,9,2,0,4,7,1,0,6,4,7
%N A060214 Successive digits of the Lucas sequence.
%H A060214 Paul Cooijmans, <a href="http://web.archive.org/web/20050302174449/http://members.chello.nl/p.cooijmans/gliaweb/tests/odds.html">Odds</a> (begins with the terms 1, 2, 3, 1, 9, 9,...)
%Y A060214 Cf. A000032.
%Y A060214 Adjacent sequences: A060211 A060212 A060213 this_sequence A060215 A060216 A060217
%Y A060214 Sequence in context: A130077 A080412 A098164 this_sequence A030133 A139374 A111958
%K A060214 easy,nonn,base
%O A060214 2,1
%A A060214 Jason Earls (zevi_35711(AT)yahoo.com), Mar 20 2001

%I A030133
%S A030133 2,1,3,4,7,2,9,2,2,4,6,1,7,8,6,5,2,7,9,7,7,5,3,8,2,1,3,4,7,2,9,2,2,
%T A030133 4,6,1,7,8,6,5,2,7,9,7,7,5,3,8,2,1,3,4,7,2,9,2,2,4,6,1,7,8,6,5,2,7,
%U A030133 9,7,7,5,3,8,2,1,3,4,7,2,9,2,2,4,6,1,7,8,6,5,2,7,9,7,7,5,3,8,2,1,3
%N A030133 a(n+1) = sum of digits of (a(n) + a(n-1)).
%F A030133 a(n+24) = a(n); a(A017593(n)) = 9. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 04 2007
%Y A030133 Cf. A030132, A007953.
%Y A030133 Adjacent sequences: A030130 A030131 A030132 this_sequence A030134 A030135 A030136
%Y A030133 Sequence in context: A080412 A098164 A060214 this_sequence A139374 A111958 A075616
%K A030133 nonn,base,nice
%O A030133 0,1
%A A030133 njas

%I A139374
%S A139374 2,1,3,4,7,2,9,11,11,13,6,19,7,8,15,14,11,16,27,25,16
%N A139374 Digit sum of Lucas numbers.
%e A139374 15127 is a Lucas number whose digit sum is 16.
%Y A139374 Cf. A000032, A004090.
%Y A139374 Adjacent sequences: A139371 A139372 A139373 this_sequence A139375 A139376 A139377
%Y A139374 Sequence in context: A098164 A060214 A030133 this_sequence A111958 A075616 A050041
%K A139374 nonn,base
%O A139374 0,1
%A A139374 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jun 08 2008

%I A111958
%S A111958 2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7,4,3,
%T A111958 7,2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7,4,
%U A111958 3,7,2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7
%N A111958 Lucas numbers (A000032) mod 8.
%D A111958 P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.
%Y A111958 Cf. A000032, A111958.
%Y A111958 Adjacent sequences: A111955 A111956 A111957 this_sequence A111959 A111960 A111961
%Y A111958 Sequence in context: A060214 A030133 A139374 this_sequence A075616 A050041 A058658
%K A111958 nonn
%O A111958 0,1
%A A111958 njas, Nov 28 2005

%I A075616
%S A075616 2,1,3,4,7,5,9,6,17,8,43,10,19,11,13,12,33,14,37,15,23,16,113,18,73,20,
%T A075616 51,21,103,22,61,24,87,25,49,26,67,27,117,28,57,29,71,30,79,31,283,32,
%U A075616 239,34,81,35,143,36,151,38,139,39,147,40,289,41,161,42,351,44,129,45
%N A075616 Rearrangement of natural numbers so that starting with a(n) the concatenation of n numbers gives a prime.
%t A075616 a = {2, 1, 3}; f[n_] := Block[{k = 1, b = Take[a, {n, 2n - 3}]}, While[Position[a, k] != {}, k++ ]; b = Join[b, IntegerDigits[k]]; a = Append[a, k]; While[ Position[a, k] != {} || !PrimeQ[ FromDigits[ Join[ b, IntegerDigits[k]]]], k++ ]; a = Append[a, k]]; Do[ f[n], {n, 3, 40}]; a
%Y A075616 Cf. A075617 and A073672.
%Y A075616 Adjacent sequences: A075613 A075614 A075615 this_sequence A075617 A075618 A075619
%Y A075616 Sequence in context: A030133 A139374 A111958 this_sequence A050041 A058658 A070827
%K A075616 base,nonn
%O A075616 1,1
%A A075616 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 29 2002
%E A075616 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 02 2002

%I A050041
%S A050041 1,2,1,3,4,7,8,10,11,21,29,36,40,43,44,46,47,93,137,180,220,256,285,
%T A050041 306,317,327,335,342,346,349,350,352,353,705,1055,1404,1750,2092,2427,
%U A050041 2754,3071,3377,3662,3918,4138,4318,4455,4548,4595
%N A050041 a(n)=a(n-1)+a(m), where m=2^(p+1)+2-n, and 2^p<n-1<=2^(p+1), for n >= 4.
%Y A050041 Adjacent sequences: A050038 A050039 A050040 this_sequence A050042 A050043 A050044
%Y A050041 Sequence in context: A139374 A111958 A075616 this_sequence A058658 A070827 A000032
%K A050041 nonn
%O A050041 1,2
%A A050041 Clark Kimberling (ck6(AT)evansville.edu)

%I A058658
%S A058658 1,2,1,3,4,7,8,13,15,21,26,35,42
%N A058658 McKay-Thompson series of class 38a for Monster.
%D A058658 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058658 T38a = 1/q + 2*q + q^3 + 3*q^5 + 4*q^7 + 7*q^9 + 8*q^11 + 13*q^13 + 15*q^15 + ...
%Y A058658 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058658 Adjacent sequences: A058655 A058656 A058657 this_sequence A058659 A058660 A058661
%Y A058658 Sequence in context: A111958 A075616 A050041 this_sequence A070827 A000032 A061084
%K A058658 nonn
%O A058658 -1,2
%A A058658 njas, Nov 27, 2000

%I A070827
%S A070827 2,1,3,4,7,11,8,29,47,23,44,199,32,521,284,46,2207,3571,118,9349,2168,
%T A070827 244,353,600,1152,263,90484,5802,14517,19548,2570,3010349,5568,10104,
%U A070827 63513,1022,103713,54018521,29134604
%N A070827 Sum of prime factors of Lucas numbers A000032(n),n=0, n>=2, with n=1 term added.
%C A070827 a(1) := 1 for A000032(1)=1. Sometimes a(1)=0 is chosen.
%Y A070827 Cf. A064725 ( for Fibonacci).
%Y A070827 Adjacent sequences: A070824 A070825 A070826 this_sequence A070828 A070829 A070830
%Y A070827 Sequence in context: A075616 A050041 A058658 this_sequence A000032 A061084 A055391
%K A070827 nonn,easy
%O A070827 0,1
%A A070827 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 10, 2002

%I A000032 M0155
%S A000032 2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,
%T A000032 9349,15127,24476,39603,64079,103682,167761,271443,439204,710647,
%U A000032 1149851,1860498,3010349,4870847,7881196,12752043,20633239,33385282
%N A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). (Cf. A000204.)
%C A000032 This is also the Horadam sequence (2,1,1,1). - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%C A000032 For distinct primes p,q, L(p) is congruent 1 mod p, L(2p) is congruent 3 mod p, and L(pq) is congruent 1+q(L(q)-1) mod p. Also, L(m) divides F(2km) and L((2k+1)m), k,m >=0.
%C A000032 a(n)=sum(P(3;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(0)=2. These are the sums over the SW-NE diagonals in P(3;n,k), the (3,1) Pascal triangle A093560. Observation by Paul Barry (pbarry(AT)wit.ie), Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums of the (1,2) Pascal triangle A029635 (with T(0,0) replaced by 2).
%C A000032 Suppose psi=ln(phi). We get the representation L(n)=2*cosh(n*psi) if n is even; L(n)=2*sinh(n*psi) if n is odd. There is a similar representation for Fibonacci numbers (A000045). Many Lucas formulas now easily follow from appropriate sinh- and cosh-formulas. For example: the identity cosh^2(x)-sinh^2(x)=1 implies L(n)^2-5F(n)^2=4*(-1)^n (setting x=n*psi). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Apr 18 2007
%C A000032 Comments from John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Oct 02 2007, Oct 11 2007: (Start) The parity of L(n) follows easily from its definition, which shows that L(n) is even when n is a multiple of 3, and odd otherwise.
%C A000032 The first six multiplication formulae are:
%C A000032 L(2n) = (L(n))^2 - 2*(-1)^n
%C A000032 L(3n) = (L(n))^3 - 3*((-1)^n)*L(n)
%C A000032 L(4n) = (L(n))^4 - 4*((-1)^n)*(L(n))^2 + 2
%C A000032 L(5n) = (L(n))^5 - 5*((-1)^n)*(L(n))^3 + 5*L(n)
%C A000032 L(6n) = (L(n))^6 - 6*((-1)^n)*(L(n))^4 + 9*(L(n))^2 - 2*(-1)^n
%C A000032 Generally, L(n) | L(mn) iff m is odd. (End)
%C A000032 In the expansion of L(mn), where m represents the multiplier and n the index of a known value of L(n), the absolute values of the coefficients are the terms in the m-th row of the triangle A034807. When m=1 and n=1, L(n)=1 and all the terms are positive, and so the row sums of A034807 are simply the Lucas numbers. (End)
%C A000032 The comments submitted by Miklos Kristof on Mar 19 2007 for the Fibonacci numbers (A00045) contain four important identities which have close analogues in the Lucas numbers: For a>=b and odd b, L(a+b) + L(a-b) = 5*F(a)*F(b). For a>=b and even b, L(a+b) + L(a-b) = L(a)*L(b). For a>=b and odd b, L(a+b) - L(a-b) = L(a)*L(b). For a>=b and even b, L(a+b) - L(a-b) = 5*F(a)*F(b). - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Nov 15 2007. A particularly interesting instance of the difference identity for even b is L(a+30) - L(a-30) = 5*F(a)*832040, since 5*832040 is divisible by 100, proving that the last two digits of Lucas nu mbers repeat in a cycle of length 60.
%C A000032 Further comments from John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Nov 15 2007: (Start) The Lucas numbers satisfy remarkable difference equations, in some cases best expressed using Fibonacci numbers, of which representative examples are the following:
%C A000032 L(n) - L(n-3) = 2*L(n-2)
%C A000032 L(n) - L(n-4) = 5*F(n-2)
%C A000032 L(n) - L(n-6) = 4*L(n-3)
%C A000032 L(n) - L(n-12) = 40*F(n-6)
%C A000032 L(n) - L(n-60) = 4160200*F(n-30).
%C A000032 These formulae establish, respectively, that the Lucas numbers form a cyclic residue system of length 3 (mod 2), of length 4 (mod 5), of length 6 (mod 4), of length 12 (mod 40), and of length 60 (mod 4160200). The divisibility of the last modulus by 100 accounts for the fact that the last two digits of the Lucas numbers begin to repeat at L(60).
%C A000032 The divisibility properties of the Lucas numbers are very complex and still not fully understood, but several important criteria are established in Zhi-Hong Sun's 2003 survey of congruences for Fibonacci numbers. (End)
%D A000032 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
%D A000032 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 32,50.
%D A000032 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
%D A000032 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
%D A000032 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%D A000032 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%D A000032 A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37.
%D A000032 Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
%D A000032 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%D A000032 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
%H A000032 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b000032.txt">The first 500 Lucas numbers: Table of n, L(n) for n = 0..500</a>
%H A000032 G. Everest, Y. Puri and T. Ward, <a href="http://arXiv.org/abs/math.NT/0204173">Integer sequences counting periodic points</a>
%H A000032 R. Javonovic, <a href="http://milan.milanovic.org/math/english/function1/function1.html">Lucas Function Calculator</a>
%H A000032 B. Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/lucas.txt">Factorizations of Lucas numbers</a>
%H A000032 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A000032 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html">The Lucas numbers</a>
%H A000032 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucas200.html">The First 200 Lucas numbers and their factors</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha113.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha114.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha115.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha116.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha117.htm">Factorizations of many number sequences</a>
%H A000032 B. Rittaud, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.pdf">On the Average Growth of Random Fibonacci Sequences</a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
%H A000032 Zhi-Hong Sun, <a href="http://202.195.112.2/xsjl/szh/ConFn.pdf">Congruences for Fibonacci Numbers</a> [PDF] (Lecture notes, 2003)
%H A000032 Dan Sewell Ward, <a href="http://www.halexandria.org/dward094.htm">Modified Fibonacci Sequence</a>.
%H A000032 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Link to a section of The World of Mathematics.</a>
%H A000032 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A000032 <a href="http://www.research.att.com/~njas/sequences/Sindx_Rea.html#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%F A000032 Conjecture: Let f(n) = Phi^n + Phi^(-n), then L(2n) = f(2n) and L(2n+1) = f(2n+1) - 2*Sum(k=0..infinity, C(k+1)/f(2n+1)^(2k+1)) where C(n) are Catalan numbers (A000108). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Dec 21 2007
%F A000032 G.f.: (2-x)/(1-x-x^2). L(n)=((1+sqrt(5))/2)^n + ((1-sqrt(5))/2)^n.
%F A000032 L(n) = L(n-1) + L(n-2) = (-1)^n L(-n).
%F A000032 E.g.f.: 2*exp(x/2)*cosh(sqrt(5)*x/2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 30 2001
%F A000032 L(n) = Fibonacci(2n)/Fibonacci(n) [ Jeff Burch (gburch(AT)erols.com) ]
%F A000032 L(n) = Fib(n) + 2*Fib(n-1) = Fib(n + 1) + Fib(n-1) - Henry Bottomley (se16(AT)btinternet.com), Apr 12 2000
%F A000032 a(n)=sqrt(F(n-1)^2+4*F(n)*F(n-2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 06 2003
%F A000032 a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)5^k}. a(n)=2T(n, i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
%F A000032 L(n)=2*Fib(n+1)-Fib(n) - Paul Barry (pbarry(AT)wit.ie), Mar 22 2004
%F A000032 a(n)=floor((phi)^n+(-phi)^(-n)) - Paul Barry (pbarry(AT)wit.ie), Mar 12 2005
%F A000032 Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Mar 19 2007: (Start)
%F A000032 Let F(n)=A000045=Fibonacci numbers, L(n)=a(n)=Lucas numbers:
%F A000032 L(n+m)+(-1)^m*L(n-m)=L(n)*L(m)
%F A000032 L(n+m)-(-1)^m*L(n-m)=8*F(n)*F(m)
%F A000032 L(n+m+k)+(-1)^k*L(n+m-k)+(-1)^m*(L(n-m+k)+(-1)^k*L(n-m-k))=L(n)*L(m)*L(k)
%F A000032 L(n+m+k)-(-1)^k*L(n+m-k)+(-1)^m*(L(n-m+k)-(-1)^k*L(n-m-k))=5*F(n)*L(m)*F(k)
%F A000032 L(n+m+k)+(-1)^k*L(n+m-k)-(-1)^m*(L(n-m+k)+(-1)^k*L(n-m-k))=5*F(n)*F(m)*L(k)
%F A000032 L(n+m+k)-(-1)^k*L(n+m-k)-(-1)^m*(L(n-m+k)-(-1)^k*L(n-m-k))=5*L(n)*F(m)*F(k) (End)
%F A000032 Inverse: floor(log_phi(a(n))+0.5)=n, for n>1. Also for n>=0, floor(1/2*log_phi(a(n)*a(n+1)))=n. Extension valid for all integers n: floor(1/2*sign(a(n)*a(n+1))*log_phi|a(n)*a(n+1)|)=n {where sign(x) = sign of x}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 02 2007
%F A000032 Starting (1, 3, 4, 7, 11,...) = row sums of triangle A131774. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007
%F A000032 a(n)=2*fibonacci(n-1)+fibonacci(n), n>=0 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%F A000032 a(n) = trace of the 2 X 2 matrix [0,1; 1,1]^n - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 02 2008
%p A000032 with(combinat): A000032 := n->fibonacci(n+1)+fibonacci(n-1);
%p A000032 a:=n->2*fibonacci(n-1)+fibonacci(n): seq(a(n), n=0..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%t A000032 a[0] := 2; a[n] := Nest[{Last[ # ], First[ # ] + Last[ # ]} &, {2, 1}, n] // Last
%t A000032 Array[2 Fibonacci[ #+1] - Fibonacci[ # ] &, 50, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006
%o A000032 (MAGMA) [ Lucas(n) : n in [0..120]];
%o A000032 (PARI) a(n)=if(n<0,(-1)^n*a(-n),if(n<2,2-n,a(n-1)+a(n-2)))
%o A000032 (PARI) a(n)=if(n<0,(-1)^n*a(-n),polsym(x^2-x-1,n)[n+1])
%o A000032 (PARI) a(n)=real((2+quadgen(5))*quadgen(5)^n)
%o A000032 sage: [lucas_number2(n,1,-1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%Y A000032 Cf. A000204. A000045(n)=(2L(n+1)-L(n))/5.
%Y A000032 First row of array A103324.
%Y A000032 a(n) = A101220(2,0,n), for n > 0.
%Y A000032 a(k) = A090888(1, k) = A109754(2, k) = A118654(2, k-1), for k > 0.
%Y A000032 Cf. A131774.
%Y A000032 Adjacent sequences: A000029 A000030 A000031 this_sequence A000033 A000034 A000035
%Y A000032 Sequence in context: A050041 A058658 A070827 this_sequence A061084 A055391 A134876
%K A000032 nonn,nice,easy,core
%O A000032 0,1
%A A000032 njas

%I A061084
%S A061084 1,2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,
%T A061084 15127,24476,39603,64079,103682,167761,271443,439204,710647,1149851,1860498,
%U A061084 3010349,4870847,7881196,12752043,20633239,33385282,54018521
%V A061084 1,2,-1,3,-4,7,-11,18,-29,47,-76,123,-199,322,-521,843,-1364,2207,-3571,5778,-9349,
%W A061084 15127,-24476,39603,-64079,103682,-167761,271443,-439204,710647,-1149851,1860498,
%X A061084 -3010349,4870847,-7881196,12752043,-20633239,33385282,-54018521
%N A061084 Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
%C A061084 If we drop 1 and start with 2 this is the reflected (definition A074058) Lucas sequence with a(0)=2, a(1)=-1. G.f.: (2+x)/(1+x-x^2). In this case a(n) is also the trace of A^(-n), where A is the Fibomatrix ((1,1), (1,0)). - Mario Catalani (mario.catalani(AT)unito.it), Aug 17 2002
%C A061084 The positive sequence with g.f. (1+x-2x^2)/(1-x-x^2) gives the diagonal sums of the Riordan array (1+2x,x/(1-x)). - Paul Barry (pbarry(AT)wit.ie), Jul 18 2005
%H A061084 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b061084.txt">Table of n, a(n) for n=0..500</a>
%H A061084 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%F A061084 a(n) = (-1)^(n-1) * ((n-1)-st Lucas number), see A000204
%F A061084 O.g.f.: (3*x+1)/(1+x-x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 02 2001
%e A061084 a(6) = a(4)-a(5) = -4 - 7 = -11
%Y A061084 Cf. A061083 for division, A000301 for multiplication and A000045 for addition - the common Fibonacci numbers
%Y A061084 Adjacent sequences: A061081 A061082 A061083 this_sequence A061085 A061086 A061087
%Y A061084 Sequence in context: A058658 A070827 A000032 this_sequence A055391 A134876 A019612
%K A061084 sign,easy,nice
%O A061084 0,2
%A A061084 Ulrich Schimke (ulrschimke(AT)aol.com)
%E A061084 Corrected by T. D. Noe (noe(AT)sspectra.com), Oct 25 2006

%I A055391
%S A055391 2,1,3,4,7,11,167761
%N A055391 Palindromic Lucas numbers.
%e A055391 11 is the sixth Lucas number, and it is also palindromic in base 10.
%Y A055391 Cf. A000032, A045504.
%Y A055391 Adjacent sequences: A055388 A055389 A055390 this_sequence A055392 A055393 A055394
%Y A055391 Sequence in context: A070827 A000032 A061084 this_sequence A134876 A019612 A007444
%K A055391 nonn,base
%O A055391 2,1
%A A055391 M. Harminc (harminc(AT)duro.science.upjs.sk), Jul 06 2000
%E A055391 Perhaps 167761 is the last term?
%E A055391 No other terms through L(10000) - James A. Sellers (sellersj(AT)math.psu.edu), Jul 07 2000
%E A055391 No further terms through L(200000). - Lior Manor (lior.manor(AT)gmail.com), Oct 18 2007

%I A134876
%S A134876 1,2,1,3,4,8,18,23,44,73,142,277,484,871,1644,3060,5851,10917,20776,
%T A134876 39263,74752,142521,271223,520242,996486,1916486,3686628,7103236,
%U A134876 13702428,26469008
%N A134876 Number of Proth primes; primes of the form 1 + k*2^n with k odd and k < 2^n.
%C A134876 All primes were found by Mathematica's PrimeQ function and proved using Proth's theorem. The ratio of consecutive terms is about 1.93.
%H A134876 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProthsTheorem.html">MathWorld: Proth's Theorem</a>
%e A134876 a(1)=1 because 3 is the only Proth prime for n=1. a(2)=2 because 5 and 13 are the only primes for n=2. a(3)=1 because 41 is the only prime for n=3.
%t A134876 Table[cnt=0; Do[If[PrimeQ[1+k*2^n], cnt++ ], {k,1,2^n,2}]; cnt, {n,20}]]
%Y A134876 Cf. A080076.
%Y A134876 Adjacent sequences: A134873 A134874 A134875 this_sequence A134877 A134878 A134879
%Y A134876 Sequence in context: A000032 A061084 A055391 this_sequence A019612 A007444 A052950
%K A134876 nonn
%O A134876 1,2
%A A134876 T. D. Noe (noe(AT)sspectra.com), Nov 17 2007

%I A019612
%S A019612 2,1,3,4,9,3,3,5,5,5,6,6,8,3,9,1,7,6,6,3,6,5,8,8,7,7,1,7,3,8,6,6,4,
%T A019612 3,6,2,3,7,5,8,7,2,2,1,3,3,9,4,1,2,7,8,7,4,0,4,6,9,9,0,0,2,8,2,5,4,
%U A019612 4,8,0,7,1,5,2,7,8,9,3,3,2,7,0,1,8,9,3,1,4,0,9,6,7,4,2,7,6,1,8,4,8
%N A019612 Decimal expansion of Pi*E/4.
%Y A019612 Adjacent sequences: A019609 A019610 A019611 this_sequence A019613 A019614 A019615
%Y A019612 Sequence in context: A061084 A055391 A134876 this_sequence A007444 A052950 A086851
%K A019612 nonn,cons
%O A019612 1,1
%A A019612 njas

%I A007444 M0156
%S A007444 2,1,3,4,9,7,15,12,18,17,29,20,39,25,33,34,57,30,65,38,53,
%T A007444 47,81,40,86,59,80,60,107,41,125,78,103,79,123,66,155,95,
%U A007444 123,90,177,75,189,110,132,115,209,100,210,114,171,134,239
%N A007444 Moebius transform of primes.
%H A007444 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%Y A007444 Adjacent sequences: A007441 A007442 A007443 this_sequence A007445 A007446 A007447
%Y A007444 Sequence in context: A055391 A134876 A019612 this_sequence A052950 A086851 A001054
%K A007444 nonn
%O A007444 1,1
%A A007444 njas

%I A052950
%S A052950 2,1,3,4,9,16,33,64,129,256,513,1024,2049,4096,8193,16384,32769,65536,
%T A052950 131073,262144,524289,1048576,2097153,4194304,8388609,16777216,
%U A052950 33554433,67108864,134217729,268435456,536870913,1073741824,2147483649
%N A052950 A simple regular expression.
%H A052950 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1009">Encyclopedia of Combinatorial Structures 1009</a>
%F A052950 G.f.: (x^3-x^2-3*x+2)/(-1+2*x)/(-1+x^2)
%F A052950 Recurrence: {a(1)=1, a(3)=4, a(2)=3, a(0)=2, -2*a(n)-a(n+1)+a(n+2)+1}
%F A052950 2^(n-1)+Sum(1/2*_alpha^(-n), _alpha=RootOf(-1+_Z^2))
%F A052950 E.g.f. : cosh(x)(1+exp(x)); a(n)=(2^n+1+(-1)^n+0^n)/2. - Paul Barry (pbarry(AT)wit.ie), Sep 18 2003
%p A052950 spec := [S,{S=Union(Sequence(Prod(Sequence(Z),Z)), Sequence(Prod(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%Y A052950 Adjacent sequences: A052947 A052948 A052949 this_sequence A052951 A052952 A052953
%Y A052950 Sequence in context: A134876 A019612 A007444 this_sequence A086851 A001054 A141487
%K A052950 easy,nonn
%O A052950 0,1
%A A052950 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052950 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 05 2000

%I A086851
%S A086851 1,1,0,2,1,3,4,10,93,8641,74666872,5575141774264374,31082205803147712138788845611865,
%T A086851 966103517589229313003894215813508352493573272034098666228778213,
%U A086851 933356006698282312572303256489816012122783406254583141248499016192865214116760703502264792586180597344027491798667186743473356
%V A086851 1,1,0,-2,1,-3,4,10,93,8641,74666872,5575141774264374,31082205803147712138788845611865,
%W A086851 966103517589229313003894215813508352493573272034098666228778213,
%X A086851 933356006698282312572303256489816012122783406254583141248499016192865214116760703502264792586180597344027491798667186743473356
%N A086851 a(0) = 1, a(n+1) = a(n)^2 - n.
%p A086851 a := proc(n) option remember: if n=0 then RETURN(1) fi: a(n-1)^2-n+1: end: for n from 0 to 15 do printf(`%d,`,a(n)) od:
%Y A086851 Adjacent sequences: A086848 A086849 A086850 this_sequence A086852 A086853 A086854
%Y A086851 Sequence in context: A019612 A007444 A052950 this_sequence A001054 A141487 A099866
%K A086851 sign,easy
%O A086851 0,4
%A A086851 David McLeod Moulton (dmoulton(AT)asianinc.org), Aug 18 2003

%I A001054
%S A001054 0,1,1,2,1,3,4,11,45,496,22319,11070225,247076351776,
%T A001054 2735190806339469599,675800965841611881515781657825,
%U A001054 1848444588685310753420392017318175868503407962176
%V A001054 0,1,-1,-2,1,-3,-4,11,-45,-496,22319,-11070225,-247076351776,
%W A001054 2735190806339469599,-675800965841611881515781657825,
%X A001054 -1848444588685310753420392017318175868503407962176
%N A001054 a(n) = a(n-1)a(n-2) - 1.
%H A001054 <a href="http://www.research.att.com/~njas/sequences/Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%Y A001054 Adjacent sequences: A001051 A001052 A001053 this_sequence A001055 A001056 A001057
%Y A001054 Sequence in context: A007444 A052950 A086851 this_sequence A141487 A099866 A085189
%K A001054 sign
%O A001054 0,4
%A A001054 R. K. Guy (rkg(AT)cpsc.ucalgary.ca)

%I A141487
%S A141487 1,2,1,3,4,13,17,82,167,1029,2594,22033,77729,874498,4050975,61638683,
%T A141487 387821570,7893076251,66271702355,1814566347320,20521356444643,
%U A141487 753627817663469,11423997447177546,563598221301212705
%N A141487 Number of n X n binary matrices, symmetric under 90 degree rotation, with no more than 1 one in any 2 X 2 subblock.
%H A141487 Ron Hardin (rhh(AT)cadence.com), Aug 09 2008, <a href="http://www.research.att.com/~njas/sequences/b141487.txt">Table of n, a(n) for n = 0..30</a>
%Y A141487 Adjacent sequences: A141484 A141485 A141486 this_sequence A141488 A141489 A141490
%Y A141487 Sequence in context: A052950 A086851 A001054 this_sequence A099866 A085189 A130466
%K A141487 nonn
%O A141487 0,2
%A A141487 Ron Hardin (rhh(AT)cadence.com), Aug 09 2008

%I A099866
%S A099866 1,2,1,3,4,20,5,35,56,504,70,770,3960,1144,1001,45045,80080,1361360,
%T A099866 204204,184756,67184,470288,323323,7436429,27457584,228813200,106234700,
%U A099866 2868336900,356948592,10351509168,145568097675,45581929575,85801279200
%N A099866 Denominator of sum of all elements M(i,j,k) = i*j/k, (i,j,k = 1..n). a(n) = Denominator[Sum[Sum[Sum[i*j/k,{i,1,n}],{j,1,n}],{k,1,n}]].
%F A099866 a(n) = Denominator[Sum[Sum[Sum[i*j/k, {i, 1, n}], {j, 1, n}], {k, 1, n}]]
%F A099866 a(n) = denominator of ((n^2+n)/2)^2*Sum(1/i, i=1..n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 30 2004
%e A099866 Sum[M(i,j,k)] begins 1, 27/2, 66, 625/3, 2055/4, 21609/20, ... So a(2) = 2.
%t A099866 Table[ Denominator[ Sum[ i*j/k, {i, n}, {j, n}, {k, n}]], {n, 33}]
%Y A099866 Corresponding numerator is A099865
%Y A099866 Cf. A099865.
%Y A099866 Adjacent sequences: A099863 A099864 A099865 this_sequence A099867 A099868 A099869
%Y A099866 Sequence in context: A086851 A001054 A141487 this_sequence A085189 A130466 A129322
%K A099866 nonn
%O A099866 1,2
%A A099866 Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 28 2004
%E A099866 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 02 2004

%I A085189
%S A085189 1,2,1,3,5,1,2,1,3,8,1,3,12,17,1,2,1,3,5,1,2,1,3,8,1,3,12,29,1,2,1,3,8,
%T A085189 1,3,12,44,1,3,12,60,77,1,2,1,3,5,1,2,1,3,8,1,3,12,17,1,2,1,3,5,1,2,1,3,
%U A085189 8,1,3,12,29,1,2,1,3,8,1,3,12,44,1,3,12,60,137,1,2,1,3,5,1,2,1,3,8,1,3
%N A085189 Terms of A085190 halved.
%H A085189 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/b085189.txt">Table of n, a(n) for n = 0..1429</a>
%F A085189 a(n) = A085190(n)/2
%Y A085189 Partial sums: A085188. Cf. A085194.
%Y A085189 Adjacent sequences: A085186 A085187 A085188 this_sequence A085190 A085191 A085192
%Y A085189 Sequence in context: A001054 A141487 A099866 this_sequence A130466 A129322 A089984
%K A085189 nonn
%O A085189 0,2
%A A085189 Antti Karttunen (my_firstname.my_surname(AT)iki.fi) Jun 14 2003

%I A130466
%S A130466 1,1,2,1,3,5,1,2,3,4,1,5,7,11,13,1,5,7,11,13,17,1,3,5,7,9,11,13,1,2,4,7,
%T A130466 8,11,13,14
%N A130466 Triangle where the n-th row consists of the n smallest positive integers which are coprime to the sum of divisors of n.
%Y A130466 Cf. A122212, A130465.
%Y A130466 Adjacent sequences: A130463 A130464 A130465 this_sequence A130467 A130468 A130469
%Y A130466 Sequence in context: A141487 A099866 A085189 this_sequence A129322 A089984 A062105
%K A130466 more,nonn,tabl
%O A130466 1,3
%A A130466 Leroy Quet (qq-quet(AT)mindspring.com), May 27 2007

%I A129322
%S A129322 1,1,2,1,3,5,1,2,4,5,1,2,3,4,6,1,3,5,7,9,11,1,2,3,4,5,6,7,1,2,4,5,8,10,
%T A129322 11,13,1,3,5,7,9,11,13,15,19,1,2,3,4,6,7,8,9,12,13,1,2,3,4,5,6,7,8,9,10,
%U A129322 11,1,5,7,11,13,17,19,23,25,29,31,35,1,2,3,4,5,6,7,8,9,10,11,12,13,1,2,3,4,5,6,7,8,9,10,11,12,14,15,1,3,7,9,11,13,17,19,21,23,27,29,31,33,37,1,2,4,5,8,10,11,13,16,17,19,20,22,23,25,26
%N A129322 Triangle where the n-th row is the smallest n positive integers which are coprime to the n-th Fibonacci number.
%e A129322 The 8th Fibonacci number is 21. So row 8 are the smallest 8 positive integers which are coprime to 21: (1,2,4,5,8,10,11,13).
%Y A129322 Cf. A120839.
%Y A129322 Adjacent sequences: A129319 A129320 A129321 this_sequence A129323 A129324 A129325
%Y A129322 Sequence in context: A099866 A085189 A130466 this_sequence A089984 A062105 A093412
%K A129322 nonn,tabl
%O A129322 1,3
%A A129322 Leroy Quet (qq-quet(AT)mindspring.com), May 26 2007
%E A129322 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 01 2007

%I A089984
%S A089984 1,1,1,1,0,1,1,1,1,0,1,1,1,1,2,1,3,5,1,2,7,3,13,97,200,2309,226573,
%T A089984 45538573,105193879657,23833987746960404,1085365814730154781188953,
%U A089984 114173840897460294190477827374165629,272121792497347519357684708535661864450
%V A089984 1,1,1,1,0,1,-1,1,-1,0,1,1,-1,1,-2,-1,3,5,-1,-2,7,-3,-13,97,200,2309,-226573,45538573,
%W A089984 -105193879657,-23833987746960404,1085365814730154781188953,
%X A089984 114173840897460294190477827374165629,272121792497347519357684708535661864450
%N A089984 1, 1, 1, 1, ... a, b, c, d, ac-bd, ...
%C A089984 Inspired by the formula for the determinant of a 2 X 2 matrix.
%C A089984 Sequence b(n,p) = a(n) (mod p), p prime, n>4, is a periodic sequence. Letting l(p) denotes the length of the period of b(n,p) is there any rule for l(p) ? - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 19 2003
%F A089984 a(1)=a(2)=a(3)=a(4)=1, for n>4 a(n)=a(n-4)*a(n-2)-a(n-3)*a(n-1).
%F A089984 a(n) is asymptotic (in absolute value) to B^(r^n) where r is the real root of 1+x^2-x^3 and B>1. - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 19 2003
%o A089984 (PARI) a=b=c=d=1;for(n=5,30,e=b*d-a*c;a=b;b=c;c=d;d=e;print1(e,","))
%Y A089984 Cf. A089983.
%Y A089984 Adjacent sequences: A089981 A089982 A089983 this_sequence A089985 A089986 A089987
%Y A089984 Sequence in context: A085189 A130466 A129322 this_sequence A062105 A093412 A119355
%K A089984 sign,easy
%O A089984 1,15
%A A089984 Ray Chandler (rayjchandler(AT)sbcglobal.net), following a suggestion of Rainer Rosenthal (r.rosenthal(AT)web.de), Nov 18 2003

%I A062105
%S A062105 1,1,2,1,3,5,1,3,8,13,1,3,9,22,35,1,3,9,26,61,96,1,3,9,27,75,171,267,1,
%T A062105 3,9,27,80,216,483,750,1,3,9,27,81,236,623,1373,2123,1,3,9,27,81,242,
%U A062105 694,1800,3923,6046,1,3,9,27,81,243,721,2038,5211,11257,17303,1,3,9,27
%N A062105 Number of ways a pawn-like piece (with the initial 2-step move forbidden, and starting from any square on the back rank) can end at various squares on infinite chess board.
%C A062105 Table formatted as a square array shows the top-left corner of the infinite board.
%H A062105 Hans L. Bodlaender, <a href="http://www.chessvariants.com/index.html">The Chess Variant Pages</a>
%H A062105 Hans L. Bodlaender et al., editors, <a href="http://www.chessvariants.com/piececlopedia.dir/index.html">The Piececlopedia (An overview of several fairy chess pieces)</a>
%p A062105 [seq(CPTVSeq(j),j=0..91)]; CPTVSeq := n -> ChessPawnTriangleV( (2+(n-((trinv(n)*(trinv(n)-1))/2))), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) );
%p A062105 ChessPawnTriangleV := proc(r,c) option remember; if(r < 2) then RETURN(0); fi; if(c < 1) then RETURN(0); fi; if(2 = r) then RETURN(1); fi; RETURN(ChessPawnTriangleV(r-1,c-1)+ChessPawnTriangleV(r-1,c)+ChessPawnTriangleV(r-1,c+1)); end;
%o A062105 (PARI) T(n,k)=if(n<1|k<1,0,if(n==1,1,T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)))
%Y A062105 A005773 gives the left column of the table. A000244 (powers of 3) gives the diagonal of the table. Variant of A062104. Cf. also A062103.
%Y A062105 Adjacent sequences: A062102 A062103 A062104 this_sequence A062106 A062107 A062108
%Y A062105 Sequence in context: A130466 A129322 A089984 this_sequence A093412 A119355 A076110
%K A062105 nonn,tabl
%O A062105 0,3
%A A062105 Antti Karttunen May 30 2001

%I A093412
%S A093412 1,2,1,3,5,1,4,7,3,1,5,3,2,7,1,6,11,5,9,4,1,7,13,3,11,5,9,1,8,5,7,13,2,
%T A093412 11,5,1,9,17,4,3,7,13,3,11,1,10,19,9,17,8,15,7,13,6,1,11,7,5,19,3,17,2,
%U A093412 5,7,13,1,12,23,11,21,10,19,9,17,8,15,7,1,13,25,6,23,11,3,5,19,9,17,4
%N A093412 Triangle read by rows: a(n, k) is the numerator of (n+(n-1)+...+(n-k+1))/(1+2+...+k), 0 < k <= n.
%C A093412 A093415 gives the corresponding denominators.
%F A093412 a(n, k) = (2n+1-k)/gcd(2n+1-k, k+1).
%Y A093412 Cf. A093413, A093414, A093415.
%Y A093412 Adjacent sequences: A093409 A093410 A093411 this_sequence A093413 A093414 A093415
%Y A093412 Sequence in context: A129322 A089984 A062105 this_sequence A119355 A076110 A117584
%K A093412 easy,nonn,tabl,frac
%O A093412 1,2
%A A093412 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 30 2004
%E A093412 Edited and extended by David Wasserman (wasserma(AT)spawar.navy.mil), Feb 01 2006

%I A119355
%S A119355 1,2,1,3,5,1,4,7,8,1,5,9,16,11,1,6,11,20,30,14,1,7,13,24,44,49,17,1,8,
%T A119355 15,28,52,88,73,20,1,9,17,32,60,103,159,102,23,1
%N A119355 Triangle generated from binomial transforms of (1; 2, 3; 3, 4, 5;...).
%F A119355 Take antidiagonals of an array generated by taking binomial transforms of (1; 2, 3; 3, 4, 5; 4, 5, 6, 7;...)
%e A119355 First few rows of the array are:
%e A119355 1, 1, 1, 1,...
%e A119355 2, 5, 8, 11,...
%e A119355 3, 7, 16, 30,...
%e A119355 4, 9, 20, 44,...
%e A119355 ...
%e A119355 First few rows of the triangle are:
%e A119355 1;
%e A119355 2, 1;
%e A119355 3, 5, 1;
%e A119355 4, 7, 8, 1;
%e A119355 5, 9, 16, 11, 1;
%e A119355 6, 11, 20, 30, 14, 1;
%e A119355 7, 13, 24, 44, 49, 17, 1;
%e A119355 ...
%e A119355 Example: The diagonal 3, 7, 16, 30, 49...(third row of the array) = binomial transform of (3, 4, 5, 0, 0, 0...).
%Y A119355 Adjacent sequences: A119352 A119353 A119354 this_sequence A119356 A119357 A119358
%Y A119355 Sequence in context: A089984 A062105 A093412 this_sequence A076110 A117584 A047997
%K A119355 nonn
%O A119355 1,2
%A A119355 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 16 2006

%I A076110
%S A076110 1,1,2,1,3,5,1,4,7,10,1,5,9,13,17,1,6,11,16,21,26,1,7,13,19,25,31,37,1,
%T A076110 8,15,22,29,36,43,50,1,9,17,25,33,41,49,57,65,1,10,19,28,37,46,55,64,
%U A076110 73,82,1,11,21,31,41,51,61,71,81,91,101,1,12,23,34,45,56,67,78,89,100
%N A076110 Triangle (read by rows) in which the n-th row contains first n terms of an A.P. with first term 1 and common difference (n-1).
%C A076110 Leading diagonal contains n^2 + 1 (A002522). Sum of the n-th row = (n+1)(n^2+2)/2 (A064808).
%e A076110 1; 1,2; 1,3,5; 1,4,7,10; 1,5,9,13,17; 1,6,11,16,21,26; 1,7,13,19,25,31,37; ...
%Y A076110 Cf. A002522, A064808, A076111.
%Y A076110 Adjacent sequences: A076107 A076108 A076109 this_sequence A076111 A076112 A076113
%Y A076110 Sequence in context: A062105 A093412 A119355 this_sequence A117584 A047997 A049069
%K A076110 easy,nonn,tabl
%O A076110 1,3
%A A076110 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 09 2002
%E A076110 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

%I A117584
%S A117584 1,1,2,1,3,5,1,4,7,12,1,5,9,17,29,1,6,11,22,41,70,1,7,13,27,53,99,169,1,
%T A117584 8,15,32,65,128,239,408,1,9,17,37,77,157,309,577,985,1,10,19,42,89,186,
%U A117584 379,746,1393,2378
%N A117584 Generalized Pellian triangle.
%C A117584 Diagonals of the triangle are composed of the infinite set of Pellian sequences. Right border = A000129. Next diagonal going to the left = A001333 starting (1, 3, 7, 17...). A048654 = (1, 4, 9,...). A048655 = (1, 5, 11,...). A048693 = (1, 6, 13...); and so on.
%F A117584 Antidiagonals of the generalized Pellian array. First row of the array = A000129: (1, 2, 5, 12...). n-th row of the array starts (1, n+1,...); as a Pellian sequence.
%e A117584 First few rows of the triangle are:
%e A117584 1;
%e A117584 1, 2;
%e A117584 1, 3, 5;
%e A117584 1, 4, 7, 12;
%e A117584 1, 5, 9, 17, 29;
%e A117584 1, 6, 11, 22, 41, 70;
%e A117584 1, 7, 13, 27, 53, 99, 169;
%e A117584 ...
%e A117584 The triangle rows are antidiagonals of the generalized Pellian array:
%e A117584 1, 2, 5, 12, 29,...
%e A117584 1, 3, 7, 17, 41,...
%e A117584 1, 4, 9, 22, 53,...
%e A117584 1, 5, 11, 27, 65,...
%e A117584 ...
%e A117584 For example, in the row (1, 5, 11, 27, 65...), 65 = 2*27 + 11.
%Y A117584 Cf. A000129, A001333, A048654, A048655, A048693.
%Y A117584 Adjacent sequences: A117581 A117582 A117583 this_sequence A117585 A117586 A117587
%Y A117584 Sequence in context: A093412 A119355 A076110 this_sequence A047997 A049069 A030237
%K A117584 nonn
%O A117584 1,3
%A A117584 Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 29 2006

%I A047997
%S A047997 1,1,2,1,3,5,1,4,8,12,1,5,13,24,32,1,6,18,43,73,94,1,7,25,69,141,227,
%T A047997 289,1,8,32,104,252,480,734,910,1,9,41,150,414,920,1656,2430,2934,1,10,
%U A047997 50,207,649,1636,3370,5744,8150,9686,1,11,61,277,967
%N A047997 Triangle of numbers a(n,k) = number of balance positions when k equal weights are placed at a k-subset of the points {-n, -(n-1), ..., n-1, n} on a centrally pivoted rod.
%D A047997 R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.
%F A047997 Equivalent to number of partitions of n(2k-n+1)/2 into up to n parts each no more than 2k-n+1 so a(n, k)=A067059(n, n(2k-n+1)/2); row sums are A047653(n)-1. - Henry Bottomley (se16(AT)btinternet.com), Aug 11 2001
%Y A047997 a(n, n) is A002838.
%Y A047997 Adjacent sequences: A047994 A047995 A047996 this_sequence A047998 A047999 A048000
%Y A047997 Sequence in context: A119355 A076110 A117584 this_sequence A049069 A030237 A118243
%K A047997 nonn,nice,tabl
%O A047997 1,3
%A A047997 njas

%I A049069
%S A049069 1,1,2,1,3,5,1,4,9,13,1,5,13,25,33,1,6,17,37,65,81,1,7,21,49,97,161,
%T A049069 193,1,8,25,61,129,241,385,449,1,9,29,73,161,321,577,897,1025,1,10,
%U A049069 33,85,193,401,769,1345,2049,2305,1,11,37,97,225,481,961,1793,3073
%N A049069 Array T by antidiagonals: T(k,n)=k*n*2^(n-1)+1, n >= 0, k >= 1.
%e A049069 Antidiagonals: {1}; {1,2}; {1,3,5}; ...
%o A049069 (PARI) T(k,n)=k*n*2^(n-1)+1
%Y A049069 Transpose of the array in A048472.
%Y A049069 Row 1 = (1, 2, 5, 13, 33, ...) = A005183.
%Y A049069 Row 2 = (1, 3, 9, 25, 65, ...) = A002064.
%Y A049069 Cf. A049513.
%Y A049069 Essentially the same as A049513.
%Y A049069 Adjacent sequences: A049066 A049067 A049068 this_sequence A049070 A049071 A049072
%Y A049069 Sequence in context: A076110 A117584 A047997 this_sequence A030237 A118243 A134081
%K A049069 nonn,tabl,easy
%O A049069 0,3
%A A049069 njas, Clark Kimberling (ck6(AT)evansville.edu), Michael Somos

%I A030237
%S A030237 1,1,2,1,3,5,1,4,9,14,1,5,14,28,42,1,6,20,48,90,132,
%T A030237 1,7,27,75,165,297,429,8,35,110,275,572,1001,1430,1,9,44,154,
%U A030237 429,1001,2002,3432,4862
%N A030237 Catalan's triangle with right border removed: 1; 1,2; 1,3,5; ...
%C A030237 This triangle appears in the totally asymmetric exclusion process as Y(alpha=1,beta=1,n,m), written in the Derrida et al. reference as Y_n(m) for alpha=1, beta=1. - Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 13 2006.
%D A030237 B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
%H A030237 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A030237.text">First 10 rows.</a>
%F A030237 m-th entry in row n is (n+m)!/n!/m! /(n+1) (n-m+1).
%Y A030237 Cf. A009766.
%Y A030237 Row sums give A071724(n)= 3*binomial(2*n, n-1)/(n+2), n>=1.
%Y A030237 The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y A030237 Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...
%Y A030237 Adjacent sequences: A030234 A030235 A030236 this_sequence A030238 A030239 A030240
%Y A030237 Sequence in context: A117584 A047997 A049069 this_sequence A118243 A134081 A134247
%K A030237 nonn,tabl
%O A030237 0,3
%A A030237 Wouter L. J. Meeussen (wouter.meeussen(AT)pandora.be).

%I A118243
%S A118243 1,1,2,1,3,5,1,4,10,12,1,5,17,33,29,1,6,26,72,109,70,1,7,37,135,305,360,
%T A118243 169,1,8,50,228,701,1292,1189,408,1,9,65,357,1405,3640,5473,3927,985,1,
%U A118243 10,82,528,2549,8658,18901,23184,12970,2378,1,11,101,747,4289
%N A118243 Triangle generated from Pell polynomials.
%C A118243 a(k)/a(k-1) of the array sequences tend to exp ArcSinh(N/2) with rows starting N = 2, 3, 4...For example terms of the Pell sequence row N=2 tend to converge to 2.414...= (1 + sqrt(2)).
%F A118243 Triangle, antidiagonals of the array in A073133, deleting the first row (Fibonacci numbers). Columns are generated as f(x) from the Pell polynomials (analogous to the Fibonacci polynomials).
%e A118243 First few rows of the triangle are:
%e A118243 1;
%e A118243 1, 2;
%e A118243 1, 3, 5;
%e A118243 1, 4, 10, 12;
%e A118243 1, 5, 17, 33, 29;
%e A118243 1, 6, 26, 72, 109, 70;
%e A118243 ...
%e A118243 Deleting first row of the A073133 array, the generating array of the triangle is
%e A118243 1, 2, 5, 12, 29,...
%e A118243 1, 3, 10, 33, 109,...
%e A118243 1, 4, 17, 72, 305,