The Database of Integer Sequences, Part 9
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 A097865
%S A097865 0,1,1,1,1,2,1,2,4,1,1,2,1,1,2,2,4,7,2,2,4,2,2,4,4,8,14,3,3,6,3,3,6,6,
%T A097865 12,21,3,3,6,3,3,6,6,12,21,4,4,8,4,4,8,8,16,28,5,5,10,5,5,10,10,20,35,5,
%U A097865 5,10,5,5,10,10,20,35,6,6,12,6,6,12,12,24,42,6,6,12,6,6,12,12,24,42,6,6
%N A097865 The n-th group (n>=0) of 9 consecutive terms are the entries, read by rows, of the 3 by 3 matrix A[n], where A[0]= [[0,1,1],[1,1,2],[1,2,4]], A[1]=[[1,1,2],[1,1,2],[2,4,7]], and A[n]=A[n-A[n-1](1,1)] + A[n-A[n-2](1,2)], (M(i,j) denotes the (i,j)-entry of the matrix M).
%C A097865 A 3 X 3 matrix Hofstadter type sequence.
%e A097865 We have A[1](1,1)=1, A[0](1,2)=1 and so A[2]=A[2-1]+A[2-1]=2A[1], a matrix with entries (2,2,4,2,2,4,4,8,14), yielding the 19th,20th,...,27th terms of the sequence.
%p A097865 with(linalg): A[0]:=matrix(3,3,[0,1,1,1,1,2,1,2,4]): A[1]:=matrix(3,3,[1,1,2,1,1,2,2,4,7]): for n from 2 to 10 do A[n]:=evalm(A[n-A[n-1][1,1]]+A[n-A[n-2][1,2]]) od: seq(seq(seq(A[k][i,j],j=1..3),i=1..3),k=0..10);
%t A097865 Clear[A] (* Hofstatder 3 X 3 Matrix sequence*) digits=50 A[n_]:=A[n]=A[n-A[n-1][[1, 1]]]+A[n-A[n-2][[1, 2]]]; A[0]:={{0, 1, 1}, {1, 1, 2}, {1, 2, 4}}; A[1]:={{1, 1, 2}, {1, 1, 2}, {2, 4, 7}}; (* flattened sequence of 3 X 3 matrices made with a Hofstadter recurrence*) b=Flatten[Table[A[n], {n, 0, digits}]] ListPlot[b, PlotJoined->True]
%Y A097865 Adjacent sequences: A097862 A097863 A097864 this_sequence A097866 A097867 A097868
%Y A097865 Sequence in context: A110330 A097864 A097866 this_sequence A105245 A105246 A132082
%K A097865 nonn
%O A097865 0,6
%A A097865 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 30 2004
%E A097865 Edited by njas, May 13 2006
%I A105245
%S A105245 0,1,1,1,1,2,1,2,4,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,2,4,
%T A105245 7,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,2,4,7,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,0,
%U A105245 1,1,1,1,2,1,2,4,2,4,7,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4
%N A105245 A 3 X 3 X 3 tensor matrix Fibonacci ( Bonacci) triangular array.
%F A105245 M={M1, M1, M2} v[n]=M.v[n-1] a(n) = v[m][[n]] by Flattening the tensor
%t A105245 v[1] = {{0, 1, 1}, {1, 1, 2}, {1, 2, 4}} M1 = {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}} M2 = {{0, 1, 0}, {0, 0, 1}, {1, 1, 1}} M = {M1, M1, M2} v[n_] := v[n] = M.v[n - 1] a = Table[v[n], {n, 1, 4}] aa = Flatten[a]
%Y A105245 Adjacent sequences: A105242 A105243 A105244 this_sequence A105246 A105247 A105248
%Y A105245 Sequence in context: A097864 A097866 A097865 this_sequence A105246 A132082 A129644
%K A105245 nonn,uned
%O A105245 1,6
%A A105245 Roger Bagula (rlbagulatftn(AT)yahoo.com), Apr 13 2005
%I A105246
%S A105246 0,1,1,1,1,2,1,2,4,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,0,1,1,2,4,7,1,2,4,1,1,
%T A105246 2,1,1,2,1,2,4,0,1,1,2,4,7,1,2,4,1,1,2,1,1,2,1,2,4,0,1,1,1,1,2,1,2,4,0,
%U A105246 1,1,2,4,7,1,2,4,1,1,2,1,1,2,1,2,4,0,1,1,4,6,11,3,6,12,1,3,4,2,4,7,1,2
%N A105246 A 3 X 3 X 3 tensor matrix Fibonacci ( Bonacci) triangular array, 2nd type/isomer.
%C A105246 The M2 matrix is taken upside down here.
%F A105246 M={M1, M1, M2} v[n]=M.v[n-1] a(n) = v[m][[n]] by Flattening the tensor
%t A105246 (* v[1] is taken as a[n]=a[n-1]+a[n-2]+a[n-3] elements*) v[1] = {{0, 1, 1}, {1, 1, 2}, {1, 2, 4}} M1 = {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}} M2 = {{1, 1, 1}, {0, 0, 1}, {0, 1, 0}} M = {M1, M1, M2} v[n_] := v[n] = M.v[n - 1] a = Table[v[n], {n, 1, 4}] aa = Flatten[a]
%Y A105246 Adjacent sequences: A105243 A105244 A105245 this_sequence A105247 A105248 A105249
%Y A105246 Sequence in context: A097866 A097865 A105245 this_sequence A132082 A129644 A081517
%K A105246 nonn,uned
%O A105246 1,6
%A A105246 Roger Bagula (rlbagulatftn(AT)yahoo.com), Apr 13 2005
%I A132082
%S A132082 1,1,2,1,2,4,1,2,3,4,1,2,4,5,7,1,2,3,4,6,7,1,2,4,5,7,8,10,1,2,3,4,5,6,7,
%T A132082 8
%N A132082 Triangle read by rows: row n contains the n smallest positive integers which are coprime to (2^n +1).
%e A132082 2^6 + 1 = 65. So row 6 contains the 6 smallest positive integers which are coprime to 65: (1,2,3,4,6,7).
%Y A132082 Cf. A132083, A131879.
%Y A132082 Adjacent sequences: A132079 A132080 A132081 this_sequence A132083 A132084 A132085
%Y A132082 Sequence in context: A097865 A105245 A105246 this_sequence A129644 A081517 A104778
%K A132082 more,nonn,tabl
%O A132082 1,3
%A A132082 Leroy Quet (qq-quet(AT)mindspring.com), Aug 09 2007
%I A129644
%S A129644 1,1,2,1,2,4,1,2,3,4,6,12,1,2,3,4,6,8,12,24,1,2,3,4,6,8,9,12,18,24,36,
%T A129644 72,1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360,
%U A129644 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,30,36,40,45,48,60,72,80,90,120
%N A129644 The concatenated list of the positive divisors of the terms of sequence A129643. Also, a(n) = A129643(n+1)/A129643(n) -1.
%p A129644 A129644 := proc(nmax) local a,b,n,a_1; a := [1] ; b := [1] ; while nops(b) < nmax do n := nops(a)+1 ; a_1 := op(-1,a) ; a := [op(a),a_1*(op(n-1,b)+1)] ; a_1 := op(-1,a) ; b := [op(b),op(numtheory[divisors](a_1))] ; od: RETURN(b) ; end: A129644(80) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 16 2007
%Y A129644 Cf. A129643.
%Y A129644 Adjacent sequences: A129641 A129642 A129643 this_sequence A129645 A129646 A129647
%Y A129644 Sequence in context: A105245 A105246 A132082 this_sequence A081517 A104778 A085683
%K A129644 nonn
%O A129644 1,3
%A A129644 Leroy Quet (qq-quet(AT)mindspring.com), Apr 25 2007
%E A129644 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 16 2007
%I A081517
%S A081517 1,1,2,1,2,4,1,2,3,5,1,2,3,4,7,1,2,3,4,5,8,1,2,3,4,5,6,8,1,2,3,4,5,6,
%T A081517 7,9,1,2,3,4,5,6,7,8,11
%N A081517 Consider the smallest number m which can be expressed as the sum of n distinct numbers coprime to m. Sequence gives triangle (read by rows) of the set of coprime numbers pertaining to m. When there is a choice, use the lexicographically earliest solution.
%e A081517 Triangle begins:
%e A081517 1;
%e A081517 1,2;
%e A081517 1,2,4;
%e A081517 1,2,3,5;
%e A081517 1,2,3,4,7;
%e A081517 1,2,3,4,5,8;
%Y A081517 Cf. A081515, A081516.
%Y A081517 Adjacent sequences: A081514 A081515 A081516 this_sequence A081518 A081519 A081520
%Y A081517 Sequence in context: A105246 A132082 A129644 this_sequence A104778 A085683 A081532
%K A081517 tabl,nonn,more
%O A081517 1,3
%A A081517 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 27 2003
%E A081517 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 23 2007
%I A104778
%S A104778 1,1,1,2,1,2,4,1,2,3,5,10,1,2,3,5,7,13,26,1,2,3,4,5,8,11,14,20,38,76,1,
%T A104778 2,3,4,5,8,10,13,14,23,33,42,60,116,232,1,2,3,4,5,5,8,11,14,17,14,24,30,
%U A104778 40
%N A104778 Table of values with shape sequence A000041 related to involutions and multinomials.
%e A104778 The 47 multinomials can be distributed as in the following triangular array:
%e A104778 1
%e A104778 9 1
%e A104778 4 6 1
%e A104778 9 2 3 1
%e A104778 1 3 2 3 1
%e A104778 divide each term by
%e A104778 1
%e A104778 3 1
%e A104778 2 3 1
%e A104778 3 2 3 1
%e A104778 1 3 2 3 1
%e A104778 yielding
%e A104778 1
%e A104778 3 1
%e A104778 2 2 1
%e A104778 3 1 1 1
%e A104778 1 1 1 1 1
%e A104778 with column sums 10 5 3 2 1
%e A104778 therefore the fourth row of the table is 1 2 3 5 10
%e A104778 The initial rows are 1; 1; 1,2; 1,2,4; 1,2,3,5,10; 1,2,3,5,7,13,26; ...
%Y A104778 Cf. A000041, A000085, A005651, A036038, A097522, A104707, A104778.
%Y A104778 A001475 and A000085 are subsequences.
%Y A104778 Adjacent sequences: A104775 A104776 A104777 this_sequence A104779 A104780 A104781
%Y A104778 Sequence in context: A132082 A129644 A081517 this_sequence A085683 A081532 A141539
%K A104778 nonn,tabf
%O A104778 0,4
%A A104778 Alford Arnold (Alford1940(AT)aol.com), Mar 24 2005
%I A085683
%S A085683 1,1,2,1,2,4,1,2,3,6,1,2,4,8,16,1,2,3,4,6,12,1,2,4,8,16,32,64,1,2,3,4,6
%N A085683 Duplicate of A081532.
%Y A085683 Adjacent sequences: A085680 A085681 A085682 this_sequence A085684 A085685 A085686
%Y A085683 Sequence in context: A129644 A081517 A104778 this_sequence A081532 A141539 A131074
%K A085683 dead
%O A085683 1,3
%I A081532
%S A081532 1,1,2,1,2,4,1,2,3,6,1,2,4,8,16,1,2,3,4,6,12,1,2,4,8,16,32,64,1,2,3,4,6,
%T A081532 8,12,24,1,2,3,4,6,9,12,18,36,1,2,3,4,6,8,12,16,24,48,1,2,4,8,16,32,64,
%U A081532 128,256,512,1024,1,2,3,4,5,6,10,12,15,20,30,60,1,2,4,8,16,32,64,128
%N A081532 Triangle read by rows: let m be smallest number with n divisors, then row n gives divisors of m.
%C A081532 Triangle: T(n,k) = k-th divisor of smallest number having exactly n divisors, 1<=k<=n.
%C A081532 T(n,1)=1, T(n,n)=A005179(n); A000005(T(n,n))=n.
%C A081532 Conjecture: T(n,3) = A023402(n-2) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 18 2003
%H A081532 Eric Weisstein's World of Mathematics, Divisor
%e A081532 1; 1,2; 1,2,4; 1,2,3,6; 1,2,4,8,16; 1,2,3,4,6,12; ...
%Y A081532 Leading diagonal is A005179. Cf. A081533.
%Y A081532 Adjacent sequences: A081529 A081530 A081531 this_sequence A081533 A081534 A081535
%Y A081532 Sequence in context: A081517 A104778 A085683 this_sequence A141539 A131074 A059268
%K A081532 nonn,tabl
%O A081532 1,3
%A A081532 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 28 2003
%E A081532 More terms from Sam Alexander (amnalexander(AT)yahoo.com), Oct 21 2003
%I A141539
%S A141539 1,1,2,1,2,4,1,2,3,8,1,2,3,5,16,1,2,3,4,8,32,1,2,3,4,6,13,64,1,2,3,4,5,
%T A141539 9,21,128,1,2,3,4,5,7,13,34,256,1,2,3,4,5,6,10,19,55,512,1,2,3,4,5,6,8,
%U A141539 14,28,89,1024,1,2,3,4,5,6,7,11,19,41,144,2048,1,2,3,4,5,6,7,9,15,26,60
%N A141539 Square array A(n,k) of numbers of length n binary words with at least k "0" between two "1" digits (n,k >= 0), read by antidiagonals.
%C A141539 A (n, k+1) = A(n, k) - A143291 (n, k).
%F A141539 G.f. of column k: x^(-k)/(1-x-x^(k+1)). A (n,k) = 2^n if k=0, else A (n,k) = n+1 if n<=k, else A (n,k) = A(n-1, k) + A(n-k-1, k).
%e A141539 A (4,3) = 6, because 6 binary words of length 4 have at least 3 "0" between two "1" digits: 0000, 0001, 0010, 0100, 1000, 1001.
%e A141539 Square array A(n,k) begins:
%e A141539 1 1 1 1 1 1 1 1 ...
%e A141539 2 2 2 2 2 2 2 2 ...
%e A141539 4 3 3 3 3 3 3 3 ...
%e A141539 8 5 4 4 4 4 4 4 ...
%e A141539 16 8 6 5 5 5 5 5 ...
%e A141539 32 13 9 7 6 6 6 6 ...
%e A141539 64 21 13 10 8 7 7 7 ...
%e A141539 128 34 19 14 11 9 8 8 ...
%p A141539 A := proc (n, k::nonnegint) option remember; if k=0 then 2^n elif n<=k and n>=0 then n+1 elif n>0 then A(n-1, k) +A(n-k-1, k) else A(n+1+k, k) -A(n+k, k) fi end; seq (seq (A(n,d-n), n=0..d), d=0..15);
%Y A141539 Cf. column k=0: A000079, k=1: A000045(n+2), k=2: A000930(n+2), A068921, A078012(n+5), k=3: A003269(n+4), A017898(n+7), k=4: A003520(n+4), A017899(n+9), k=5: A005708(n+5), A017900(n+11), k=6: A005709(n+6), A017901(n+13), k=7: A005710(n+7), A017902(n+15), k=8: A005711(n+7), A017903(n+17), k=9: A017904(n+19), k=10: A017905(n+21), k=11: A017906(n+23), k=12: A017907(n+25), k=13: A017908(n+27), k=14: A017909(n+29). Diagonal: A000027. See also A143291.
%Y A141539 Adjacent sequences: A141536 A141537 A141538 this_sequence A141540 A141541 A141542
%Y A141539 Sequence in context: A104778 A085683 A081532 this_sequence A131074 A059268 A123937
%K A141539 nonn,tabl
%O A141539 0,3
%A A141539 Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 15 2008
%I A131074
%S A131074 1,1,2,1,2,4,1,2,4,8,0,1,3,7,15,0,0,1,4,11,26,0,0,0,1,5,16,42,0,0,0,0,1,
%T A131074 6,22,64,1,1,1,1,1,2,8,30,94,1,2,3,4,5,6,8,16,46,140,1,2,4,7,11,16,22,
%U A131074 30,46,92,232,1,2,4,8,15,26,42,64,94,140,232,464,0,1,3,7,15,30,56,98
%N A131074 Triangular array T read by rows: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.
%C A131074 All columns are periodic with period length 8. The (4+8*i)-th row equals the first (4+8*i) terms of the main diagonal (i >= 0). Main diagonal and eighth subdiagonal agree; generally j-th subdiagonal equals (j+8)-th subdiagonal.
%e A131074 First seven rows of T are
%e A131074 [ 1 ]
%e A131074 [ 1, 2 ]
%e A131074 [ 1, 2, 4 ]
%e A131074 [ 1, 2, 4, 8 ]
%e A131074 [ 0, 1, 3, 7, 15 ]
%e A131074 [ 0, 0, 1, 4, 11, 26 ]
%e A131074 [ 0, 0, 0, 1, 5, 16, 42 ].
%o A131074 (PARI) {m=13; M=matrix(m, m); for(j=1, m, M[j, 1]=if((j-1)%8<4, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, for(k=1, j, print1(M[j, k], ",")))}
%o A131074 (MAGMA) m:=13; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 8 lt 4 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; &cat[ [ M[j, k]: k in [1..j] ]: j in [1..m] ];
%Y A131074 Cf. A131022, A129961 (main diagonal of T), A131075 (first subdiagonal of T), A131076 (row sums of T), A131077 (antidiagonal sums of T). First through sixth column of T are in A131078, A131079, A131080, A131081, A131082, A131083 resp.
%Y A131074 Adjacent sequences: A131071 A131072 A131073 this_sequence A131075 A131076 A131077
%Y A131074 Sequence in context: A085683 A081532 A141539 this_sequence A059268 A123937 A138882
%K A131074 nonn,tabl
%O A131074 1,3
%A A131074 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), following a suggestion of Paul Curtz (bpcrtz(AT)free.fr), Jun 14 2007
%I A059268
%S A059268 1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,1,2,4,8,16,32,1,2,4,8,16,32,64,
%T A059268 1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128,256,1,2,4,8,16,32,64,
%U A059268 128,256,512,1,2,4,8,16,32,64,128,256,512,1024,1,2,4,8,16,32,64
%N A059268 Concatenate subsequences [2^0, 2^1, ..., 2^n] for n = 0, 1, 2, ...
%C A059268 Triangular array T(n,k) read by rows, where T(n,k) = i!*j! times coefficient of x^n*y^k in exp(x+2y).
%H A059268 J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
%F A059268 E.g.f.: exp(x+2*y) (T coordinates).
%Y A059268 Adjacent sequences: A059265 A059266 A059267 this_sequence A059269 A059270 A059271
%Y A059268 Sequence in context: A081532 A141539 A131074 this_sequence A123937 A138882 A074634
%K A059268 nonn,tabl
%O A059268 0,3
%A A059268 njas, Jan 23 2001
%I A123937
%S A123937 1,1,2,1,2,4,1,2,4,8,1,4,4,8,16,1,4,12,8,16,32,1,4,12,32,16,32,64,1,
%T A123937 4,12,32,80,32,64,128,1,4,16,32,80,192,64,128,256,1,6,16,56,80,192,
%U A123937 448,128,256,512,1,6,24,56,176,192,448,1024,256,512,1024
%N A123937 Triangle read by rows: T(x, y) = 0 if y > x, = 1 if y = 0, or = 2*Sum_{k >= 1, x-k^2 >= y} T(x-k^2, y-1) otherwise. The zeros are omitted from the sequence.
%C A123937 Comments from R. J. Mathar, Oct 31 2006:
%C A123937 This sequence provides the seeds for the construction of columns (vertical recurrence)
%C A123937 of A122510 insofar as each row of A123937 provides two sides of auxiliary arrays
%C A123937 b(.,.,.) from which a column of A122510 emerges as the third side:
%C A123937 A122510(d,n)=b(0,d,n) [with an auxiliary, virtual A122510(0,n)=1].
%C A123937 Seeds to construct two sides of b(.,.,.):
%C A123937 b(x,0,n)=A123937(n,x) for x<=n; b(n,y,n)=A123937(n,n) for y>=0.
%C A123937 Recurrence within the b(.,.,.) : b(x,y,n)=b(x,y-1,n)+b(x+1,y-1,n) for xDeterminacion geometrica de los numeros primos y perfectos".
%e A138882 Triangle begins:
%e A138882 1, 2
%e A138882 1, 2, 4
%e A138882 1, 2, 4, 8, 16
%e A138882 1, 2, 4, 8, 16, 32, 64
%e A138882 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096
%e A138882 ==============================================================
%e A138882 ..... Mersenne ..............................................
%e A138882 ....... prime ...............................................
%e A138882 n ... A000668(n) = Sum of divisors of A061652(n) .............
%e A138882 ==============================================================
%e A138882 1 ........ 3 ... = 1+2
%e A138882 2 ........ 7 ... = 1+2+4
%e A138882 3 ....... 31 ... = 1+2+4+8+16
%e A138882 4 ...... 127 ... = 1+2+4+8+16+32+64
%e A138882 5 ..... 8191 ... = 1+2+4+8+16+32+64+128+256+512+1024+2048+4096
%Y A138882 Cf. A000005, A000043, A000203, A000668, A019279, A061652, A133031.
%Y A138882 Adjacent sequences: A138879 A138880 A138881 this_sequence A138883 A138884 A138885
%Y A138882 Sequence in context: A131074 A059268 A123937 this_sequence A074634 A035015 A114791
%K A138882 nonn,tabf
%O A138882 1,2
%A A138882 Omar E. Pol (info(AT)polprimos.com), Apr 11 2008
%I A074634
%S A074634 1,1,2,1,2,4,1,2,4,9,1,2,4,9,6,1,2,4,9,6,25,1,2,4,9,6,18,10,1,2,4,9,6,
%T A074634 25,10,15,1,2,4,9,6,25,10,15,12,1,2,4,9,6,25,10,15,12,21,1,2,4,9,6,24,
%U A074634 10,15,12,21,45,1,2,4,9,6,25,10,15,12,21,30,35,1,2,4,9,6,25,10,15,12
%N A074634 Cototient-remainder triangle: triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[Cototient[x],n]=k.
%F A074634 Min{x; Mod[x-Phi[x], n]=r}, r=1..n, n=1, ...
%e A074634 1; 1,2; 1,2,4; 1,2,4,9; 1,2,4,9,6; 1,2,4,9,6,25; 1,2,4,9,6,18,10; 1,2,4,9,6,25,10,15; 1,2,4,9,6,25,10,15,12, ...
%t A074634 {k=0, s=0, fl=1}; Table[Print["#"]; Table[fl=1; Print[{r, m}]; Do[s=Mod[n-EulerPhi[n], m]; If[(s==r)&&(fl==1), Print[n]; fl=0], {n, 1, 500}], {r, 0, m-1}], {m, 1, 50}]
%Y A074634 Cf. A051953, A074625.
%Y A074634 Adjacent sequences: A074631 A074632 A074633 this_sequence A074635 A074636 A074637
%Y A074634 Sequence in context: A059268 A123937 A138882 this_sequence A035015 A114791 A129994
%K A074634 nonn,tabl
%O A074634 1,3
%A A074634 Labos E. (labos(AT)ana.sote.hu), Aug 29 2002
%I A035015
%S A035015 1,2,1,2,4,1,2,5,4,2,1,6,6,6,4,1,5,2,8,4,4,2,1,2,2,3,2,10,12,4,2,5,4,6,
%T A035015 7,6,11,4,1,2,10,8,6,8,7,5,6,4,4,1,2,5,10,2,5,8,10,16,4,11,1,2,12,2,9,
%U A035015 6,15,2,6,9,6,10,10,4,1,2,12,10,3,6,16,14,9,4,18,4,4,2,1,2,9,20,10,4
%N A035015 Period of continued fraction for square root of n-th square-free integer.
%H A035015 S. R. Finch, Class number theory
%H A035015 R. Knott, An Introduction to Continued Fractions
%e A035015 a(1)=1 because 2 is smallest square-free integer and sqrt 2 = [ 1,2,2,2,2... ] thus has an eventual period of 1.
%Y A035015 Cf. A003285, A013943.
%Y A035015 Adjacent sequences: A035012 A035013 A035014 this_sequence A035016 A035017 A035018
%Y A035015 Sequence in context: A123937 A138882 A074634 this_sequence A114791 A129994 A080246
%K A035015 nonn,easy,nice
%O A035015 0,2
%A A035015 David L. Treumann (alewifepurswest(AT)yahoo.com)
%E A035015 Corrected and extended by James A. Sellers (sellersj(AT)math.psu.edu)
%I A114791
%S A114791 1,1,2,1,2,4,1,2,6,6,1,2,7,10,11,1,2,7,13,22,18,1,2,7,14,30,42,31,1,2,7,
%T A114791 15,34,60,82,54,1,2,7,15,36,71,125,157,97,1,2,7,15,37,77,152,256,304,
%U A114791 172,1,2,7,15,37,81,168,325,513,589,309,1,2,7,15,37,83,177,367,669,1049
%N A114791 Consider the array T where the element at T(r,c) is the number of n-almost primes less than or equal to 2^m. Sequence read by successive columns beginning at r=c.
%C A114791 Eventually each column reads 1,2,7,15,37,84,187,421,914,... = A052130.
%F A114791 Each column sums to 2^n-1.
%e A114791 1 2 4 6 11 18 31 54 97 172 309 564 1028 1900 3512 6542
%e A114791 ..1 2 6 10 22 42 82 157 304 589 1124 2186 4192 8110 15658
%e A114791 ....1 2 7 13 30 60 125 256 513 1049 2082 4214 8401 16771
%e A114791 ......1 2 7 14 34 71 152 325 669 1405 2866 5931 12139
%e A114791 ........1 2 7 15 36 77 168 367 770 1643 3410 7150
%e A114791 ..........1 2 7 15 37 81 177 392 831 1790 3757
%e A114791 ............1 2 7 15 37 83 182 406 867 1880
%e A114791 ..............1 2 7 15 37 84 185 414 887
%e A114791 ................1 2 7 15 37 84 186 418
%t A114791 AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; Eric Weisstein (eww(AT)wolfram.com) Feb 07 2006
%t A114791 Table[ AlmostPrimePi[n, 2^k], {k, 12}, {n, k, 1, -1}] // Flatten
%Y A114791 First row is A007053, Cf. A052130.
%Y A114791 Adjacent sequences: A114788 A114789 A114790 this_sequence A114792 A114793 A114794
%Y A114791 Sequence in context: A138882 A074634 A035015 this_sequence A129994 A080246 A113413
%K A114791 nonn,tabl
%O A114791 1,3
%A A114791 Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 21 2006
%I A129994
%S A129994 1,2,1,2,4,1,2,6,6,1,2,8,12,8,1,2,10,20,20,10,1,2,12,30,40,30,12,1,2,14,
%T A129994 42,70,70,42,14,1,2,16,56,112,140,112,56,16,1,2,18,72,168,252,252,168,
%U A129994 72,18,1
%N A129994 Traiangle read by rows: A007318^(-1) * A131109, or equally 2*A007318 - I.
%C A129994 Row sums = (1, 3, 7, 15, 31,...).
%F A129994 Inverse binomial transform of triangle A131109. Let Pascal's triangle A007318 = P, then this is (1/P) * (2P^2 - P) = 2*P - I, I = Identity matrix.
%e A129994 First few rows of the triangle are:
%e A129994 1;
%e A129994 2, 1;
%e A129994 2, 4, 1;
%e A129994 2, 6, 6, 1;
%e A129994 2, 8, 12, 8, 1;
%e A129994 2, 10, 20, 20, 10, 1;
%e A129994 2, 12, 30, 40, 30, 12, 1;
%e A129994 ...
%Y A129994 Cf. A131109, A007318.
%Y A129994 Adjacent sequences: A129991 A129992 A129993 this_sequence A129995 A129996 A129997
%Y A129994 Sequence in context: A074634 A035015 A114791 this_sequence A080246 A113413 A125694
%K A129994 nonn,tabl
%O A129994 0,2
%A A129994 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2007
%I A080246
%S A080246 1,2,1,2,4,1,2,8,6,1,2,12,18,8,1,2,16,38,32,10,1,2,20,66,88,50,12,1,2,
%T A080246 24,102,192,170,72,14,1,2,28,146,360,450,292,98,16,1,2,32,198,608,1002,
%U A080246 912,462,128,18,1,2,36,258,952,1970,2364
%V A080246 1,-2,1,2,-4,1,-2,8,-6,1,2,-12,18,-8,1,-2,16,-38,32,-10,1,2,-20,66,-88,50,-12,1,-2,24,
%W A080246 -102,192,-170,72,-14,1,2,-28,146,-360,450,-292,98,-16,1,-2,32,-198,608,-1002,912,-462,
%X A080246 128,-18,1,2,-36,258,-952,1970,-2364
%N A080246 Signed version of A035607.
%C A080246 Written as lower triangular matrix this has inverse A080247. Row sums are (1,-1,-1,1,1,-1,-1,1,1,...) Diagonal sums are signed tribonacci numbers A078042
%F A080246 Columns are generated by (1-x)^k/(1+x)^k
%e A080246 Rows are {1}, {-2,1}, {2,-4,1}, {-2,8,-6,1}, ...
%Y A080246 Cf. A035607, A080247.
%Y A080246 Adjacent sequences: A080243 A080244 A080245 this_sequence A080247 A080248 A080249
%Y A080246 Sequence in context: A035015 A114791 A129994 this_sequence A113413 A125694 A136678
%K A080246 easy,sign,tabl
%O A080246 0,2
%A A080246 Paul Barry (pbarry(AT)wit.ie), Feb 15 2003
%I A113413
%S A113413 1,2,1,2,4,1,2,8,6,1,2,12,18,8,1,2,16,38,32,10,1,2,20,66,88,50,12,1,2,
%T A113413 24,102,192,170,72,14,1,2,28,146,360,450,292,98,16,1,2,32,198,608,1002,
%U A113413 912,462,128,18,1,2,36,258,952,1970,2364,1666,688,162,20,1,2,40,326
%N A113413 A Riordan array of coordination sequences.
%C A113413 Columns include A040000,A008574,A005899,A008412,A008413,A008414. Row sums are A078057(n)=A001333(n+1). Diagonal sums are A001590(n+3). Reverse of A035607. Signed version is A080246. Inverse is A080245.
%C A113413 For another version see A122542. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 15 2006
%F A113413 Riordan array ((1+x)/(1-x), x(1+x)/(1-x)); T(n, k)=sum{i=0..n-k, C(k+1, i)C(n-i, k)}; T(n, k)=sum{j=0..n-k, C(k+j, j)C(k+1, n-k-j)}; T(n, k)=D(n, k)+D(n-1, k) where D(n, k)=sum{j=0..n-k, C(n-k, j)C(k, j)2^j}=A008288(n, k); T(n, k)=T(n-1, k)+T(n-1, k-1)+T(n-2, k-1);
%F A113413 T(n, k)=sum{j=0..n, C(floor((n+j)/2), k)C(k, floor((n-j)/2))}; - Paul Barry (pbarry(AT)wit.ie), Nov 13 2005
%e A113413 Triangle begins
%e A113413 1;
%e A113413 2, 1;
%e A113413 2, 4, 1;
%e A113413 2, 8, 6, 1;
%e A113413 2, 12, 18, 8, 1;
%e A113413 2, 16, 38, 32, 10, 1;
%e A113413 2, 20, 66, 88, 50, 12, 1;
%Y A113413 Adjacent sequences: A113410 A113411 A113412 this_sequence A113414 A113415 A113416
%Y A113413 Sequence in context: A114791 A129994 A080246 this_sequence A125694 A136678 A110162
%K A113413 easy,nonn,tabl
%O A113413 0,2
%A A113413 Paul Barry (pbarry(AT)wit.ie), Oct 29 2005
%I A125694
%S A125694 1,2,1,2,4,1,2,8,6,1,10,4,18,8,1,6,24,26,32,10,1,42,60,18,72,50,12,1,
%T A125694 102,24,162,48,150,72,14,1,82,388,214,248,230,268,98,16,1,782,536,546,
%U A125694 800,158,600
%V A125694 1,-2,1,2,-4,1,2,8,-6,1,-10,-4,18,-8,1,6,-24,-26,32,-10,1,42,60,-18,-72,50,-12,1,-102,
%W A125694 24,162,48,-150,72,-14,1,-82,-388,-214,248,230,-268,98,-16,1,782,536,-546,-800,158,600
%N A125694 Riordan array ((1+3x-sqrt(1+2x+9x^2))/(2x),(1+3x-sqrt(1+2x+9x^2))/2).
%C A125694 First column is A125695. Row sums are A091593. Inverse of A125693.
%e A125694 Triangle begins
%e A125694 1,
%e A125694 -2, 1,
%e A125694 2, -4, 1,
%e A125694 2, 8, -6, 1,
%e A125694 -10, -4, 18, -8, 1,
%e A125694 6, -24, -26, 32, -10, 1,
%e A125694 42, 60, -18, -72, 50, -12, 1
%Y A125694 Adjacent sequences: A125691 A125692 A125693 this_sequence A125695 A125696 A125697
%Y A125694 Sequence in context: A129994 A080246 A113413 this_sequence A136678 A110162 A087704
%K A125694 easy,sign,tabl
%O A125694 0,2
%A A125694 Paul Barry (pbarry(AT)wit.ie), Nov 30 2006
%I A136678
%S A136678 1,2,1,2,4,1,2,9,6,1,1,16,20,8,1,0,24,50,35,10,1,2,32,104,112,54,12,1,4,
%T A136678 38,190,293,210,77,14,1,7,40,314,664,659,352,104,16,1,10,35,478,1349,
%U A136678 1772,1286,546,135,18,1,14,20,677,2512,4234,3992,2274,800,170,20,1
%V A136678 1,2,-1,2,-4,1,2,-9,6,-1,1,-16,20,-8,1,0,-24,50,-35,10,-1,-2,-32,104,-112,54,-12,1,-4,
%W A136678 -38,190,-293,210,-77,14,-1,-7,-40,314,-664,659,-352,104,-16,1,-10,-35,478,-1349,1772,
%X A136678 -1286,546,-135,18,-1,-14,-20,677,-2512,4234,-3992,2274,-800,170,-20,1
%N A136678 Triangular sequence based on matrices of an F_n Cartan-type ( exceptional group) with example: M(4)= {{2, -1, 0, 0}, {-1, 2, -1, 0}, {0, -2, 2, -1}, {0, 0, -1, 2}}.
%C A136678 Row sums are:
%C A136678 {1, 1, -1, -2, -2, 0, 1, 1, -1, -2, -2}
%C A136678 No polynomial recursion was found, but the Gram-Schmidt analysis
%C A136678 shows that they are probably orthogonal to each other.
%F A136678 T(n, m, d)= If[ n == m, 2, If[n == Floor[d/2] + 1 && m == Floor[d/2], -2, If[(n == m - 1 || n == m + 1), -1, 0]]]
%e A136678 {1},
%e A136678 {2, -1},
%e A136678 {2, -4, 1},
%e A136678 {2, -9, 6, -1},
%e A136678 {1, -16, 20, -8,1},
%e A136678 {0, -24, 50, -35, 10, -1},
%e A136678 {-2, -32, 104, -112, 54, -12, 1},
%e A136678 {-4, -38, 190, -293, 210, -77, 14, -1},
%e A136678 {-7, -40, 314, -664, 659, -352, 104, -16, 1},
%e A136678 {-10, -35, 478, -1349, 1772, -1286, 546, -135, 18, -1},
%e A136678 {-14, -20,677, -2512, 4234, -3992, 2274, -800, 170, -20, 1}
%t A136678 T[n_, m_, d_] := If[ n == m, 2, If[n == Floor[d/2] + 1 && m == Floor[d/2], -2, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a0 = Table[M[d], {d, 1, 10}]; Table[Det[M[d]], {d, 1, 10}]; g = Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] MatrixForm[a];
%Y A136678 Adjacent sequences: A136675 A136676 A136677 this_sequence A136679 A136680 A136681
%Y A136678 Sequence in context: A080246 A113413 A125694 this_sequence A110162 A087704 A119765
%K A136678 uned,tabl,sign
%O A136678 1,2
%A A136678 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 05 2008
%I A110162
%S A110162 1,2,1,2,4,1,2,9,6,1,2,16,20,8,1,2,25,50,35,10,1,2,36,105,112,54,12,1,2,
%T A110162 49,196,294,210,77,14,1,2,64,336,672,660,352,104,16,1,2,81,540,1386,
%U A110162 1782,1287,546,135,18,1,2,100,825,2640,4290,4004,2275,800,170,20,1
%V A110162 1,-2,1,2,-4,1,-2,9,-6,1,2,-16,20,-8,1,-2,25,-50,35,-10,1,2,-36,105,-112,54,-12,1,-2,
%W A110162 49,-196,294,-210,77,-14,1,2,-64,336,-672,660,-352,104,-16,1,-2,81,-540,1386,-1782,
%X A110162 1287,-546,135,-18,1,2,-100,825,-2640,4290,-4004,2275,-800,170,-20,1
%N A110162 Riordan array ((1-x)/(1+x),x/(1+x)^2).
%C A110162 Inverse of Riordan array A094527. Rows sums are A099837. Diagonal sums are A110164. Product of Riordan array A102587 and inverse binomial transform (1/(1+x),x/(1+x)).
%C A110162 Coefficients of polynomials related to Cartan matrices of types C_n and B_n: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2), with p(x,0) = 1; p(x,1) = 2-x; p(x,2) = x^2-4*x-2. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2008
%F A110162 T(n,k)=C(n+k,n-k)(-1)^(n-k)-C(n+k-1,n-k-1)(-1)^(n-k-1); - Paul Barry (pbarry(AT)wit.ie), Mar 22 2007
%e A110162 Rows begin
%e A110162 1;
%e A110162 -2,1;
%e A110162 2,-4,1;
%e A110162 -2,9,-6,1;
%e A110162 2,-16,20,-8,1;
%e A110162 -2,25,-50,35,-10,1;
%Y A110162 Cf. A128411. See A127677 for an almost identical triangle.
%Y A110162 Cf. A136674, A053122.
%Y A110162 Adjacent sequences: A110159 A110160 A110161 this_sequence A110163 A110164 A110165
%Y A110162 Sequence in context: A113413 A125694 A136678 this_sequence A087704 A119765 A077901
%K A110162 easy,sign,tabl
%O A110162 0,2
%A A110162 Paul Barry (pbarry(AT)wit.ie), Jul 14 2005
%I A087704
%S A087704 2,1,2,4,1,3,3,1,9,2,1,2,4,1,8,5,1,3,2,1,2,3,1,9,7,1,4,2,1,2,5,1,3,3,1,
%T A087704 4,2,1,2,8,1,6,4,1,3,2,1,2,3,1,5,4,1,6,2,1,2,7,1,3,3,1,6,2,1,2,7,1,4,5,
%U A087704 1,3,2,1,2,3,1,4,7,1,10,2,1,2,4,1,3,3,1,5,2,1,2,4,1,8,6,1,3
%N A087704 Number of steps for iteration of map x -> (5/3)*floor(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.
%C A087704 It is conjectured that an integer is always reached.
%H A087704 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
%p A087704 f2 := proc(x,y) x*floor(y); end; r := 5/3; h := proc(x) local n,y; global r; y := f2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := f2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(h(n)[2],n=2..60)];
%Y A087704 Cf. A087705, A087706, A087707.
%Y A087704 Adjacent sequences: A087701 A087702 A087703 this_sequence A087705 A087706 A087707
%Y A087704 Sequence in context: A125694 A136678 A110162 this_sequence A119765 A077901 A105619
%K A087704 nonn
%O A087704 2,1
%A A087704 njas, Sep 29 2003
%I A119765
%S A119765 1,1,1,2,1,2,4,1,7,1,3,9,1,2,3,4,6,12,1,2,3,6,9,18,1,2,3,4,6,8,12,24,1,
%T A119765 2,4,8,16,32,1,2,19,38,1,2,3,6,7,14,21,42,1,2,5,10,25,50,1,2,4,7,8,14,
%U A119765 28,56,1,2,4,8,16,32,64,1,71,1,73,1,3,5,15,25,75,1,3,9,27,81,1,2,43,86
%N A119765 Irregular array where row n is the positive integers which divide the number of terms in all previous rows. a(1)=1.
%e A119765 Array begins:
%e A119765 1
%e A119765 1
%e A119765 1,2
%e A119765 1,2,4
%e A119765 1,7
%e A119765 1,3,9
%e A119765 12 terms make up these 6 rows. So row 7 is the divisors of 12, (1,2,3,4,6,12).
%p A119765 A119765 := proc(nmax) local a,dvs; a := [1] ; while nops(a) < nmax do dvs := numtheory[divisors](nops(a)) ; a := [op(a),op(dvs) ] ; od ; end: a := A119765(300) ; for i from 1 to nops(a) do printf("%d,",a[i]) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 23 2006
%o A119765 ;;PLT DrScheme - Joshua Zucker (joshua.zucker@stanfordalumni.org), Jun 21 2006
%o A119765 ;;positive-divisors gives the list of divisors of n in decreasing order
%o A119765 (define (A119765 n seq)
%o A119765 (cond
%o A119765 [(= n 0) seq]
%o A119765 [else (A119765 (sub1 n) (append seq (reverse (positive-divisors (length seq)))))]))
%o A119765 (A119765 30 (list 1))
%Y A119765 Adjacent sequences: A119762 A119763 A119764 this_sequence A119766 A119767 A119768
%Y A119765 Sequence in context: A136678 A110162 A087704 this_sequence A077901 A105619 A121439
%K A119765 nonn,tabf
%O A119765 1,4
%A A119765 Leroy Quet (qq-quet(AT)mindspring.com), Jun 18 2006
%E A119765 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org) and R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 21 2006
%I A077901
%S A077901 1,0,2,1,2,4,1,8,2,9,10,4,25,0,34,17,18,68,15,120,2,89,154,60,393,144,418,
%T A077901 225,94,1156,799,1768,254,425,2858,2940,6649,3872,4642,4785,7886,21956,
%U A077901 20271,26456,2814,11271,61370,78268,117097,72624,33186,128385,240446,435204
%V A077901 1,0,2,1,2,4,1,8,2,9,10,4,25,0,34,17,18,68,-15,120,2,89,154,-60,393,-144,418,
%W A077901 225,-94,1156,-799,1768,-254,425,2858,-2940,6649,-3872,4642,4785,-7886,21956,
%X A077901 -20271,26456,-2814,-11271,61370,-78268,117097,-72624,33186,128385,-240446,435204
%N A077901 Expansion of (1-x)^(-1)/(1+x-x^2-2*x^3).
%Y A077901 First differences are in A077971.
%Y A077901 Adjacent sequences: A077898 A077899 A077900 this_sequence A077902 A077903 A077904
%Y A077901 Sequence in context: A110162 A087704 A119765 this_sequence A105619 A121439 A009205
%K A077901 sign
%O A077901 0,3
%A A077901 njas, Nov 17 2002
%I A105619
%S A105619 1,2,1,2,4,1,10,2,6,1,74,10,2,8,1,706,74,10,2,10,1,8162,706,74,10,2,12,
%T A105619 1,110410,8162,706,74,10,2,14,1,1708394,110410,8162,706,74,10,2,16,1,
%U A105619 29752066,1708394,110410,8162,706,74,10,2,18,1
%V A105619 1,-2,1,-2,-4,1,-10,-2,-6,1,-74,-10,-2,-8,1,-706,-74,-10,-2,-10,1,-8162,-706,-74,-10,
%W A105619 -2,-12,1,-110410,-8162,-706,-74,-10,-2,-14,1,-1708394,-110410,-8162,-706,-74,-10,-2,
%X A105619 -16,1,-29752066,-1708394,-110410,-8162,-706,-74,-10,-2,-18,1
%N A105619 Matrix inverse of triangle A105615.
%C A105619 Except for the initial few terms, all columns are equal to negative A000698 (related to double factorials).
%e A105619 Triangle begins:
%e A105619 1;
%e A105619 -2,1;
%e A105619 -2,-4,1;
%e A105619 -10,-2,-6,1;
%e A105619 -74,-10,-2,-8,1;
%e A105619 -706,-74,-10,-2,-10,1;
%e A105619 -8162,-706,-74,-10,-2,-12,1;
%e A105619 -110410,-8162,-706,-74,-10,-2,-14,1;
%e A105619 -1708394,-110410,-8162,-706,-74,-10,-2,-16,1; ...
%o A105619 (PARI) {T(n,k)=if(n=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))[n+1,k+1])}
%Y A105619 Cf. A105615, A000698, A105620 (matrix square-root).
%Y A105619 Adjacent sequences: A105616 A105617 A105618 this_sequence A105620 A105621 A105622
%Y A105619 Sequence in context: A087704 A119765 A077901 this_sequence A121439 A009205 A086754
%K A105619 sign,tabl
%O A105619 0,2
%A A105619 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2005
%I A121439
%S A121439 1,2,1,2,4,1,14,0,7,1,143,22,11,11,1,1928,260,40,40,16,1,32219,3894,385,
%T A121439 121,99,22,1,640784,70644,6496,406,406,203,29,1,14753528,1502940,128723,
%U A121439 9583,259,1184,370,37,1,385500056,36631962,2947266,205620,14076,3657,2967,621
%V A121439 1,-2,1,-2,-4,1,-14,0,-7,1,-143,-22,11,-11,1,-1928,-260,-40,40,-16,1,-32219,-3894,-385,
%W A121439 -121,99,-22,1,-640784,-70644,-6496,-406,-406,203,-29,1,-14753528,-1502940,-128723,
%X A121439 -9583,259,-1184,370,-37,1,-385500056,-36631962,-2947266,-205620,-14076,3657,-2967,621
%N A121439 Matrix inverse of triangle A121334, where A121334(n,k) = C( n*(n+1)/2 + n-k, n-k) for n>=k>=0.
%C A121439 A triangle having similar properties and complementary construction is the dual triangle A121435.
%F A121439 T(n,k) = [A121412^(-n*(n+1)/2 - 1)](n,k) for n>=k>=0; i.e., row n of A121334^-1 equals row n of matrix power A121412^(-n*(n+1)/2 - 1).
%e A121439 Triangle, A121334^-1, begins:
%e A121439 1;
%e A121439 -2, 1;
%e A121439 -2, -4, 1;
%e A121439 -14, 0, -7, 1;
%e A121439 -143, -22, 11, -11, 1;
%e A121439 -1928, -260, -40, 40, -16, 1;
%e A121439 -32219, -3894, -385, -121, 99, -22, 1;
%e A121439 -640784, -70644, -6496, -406, -406, 203, -29, 1;
%e A121439 -14753528, -1502940, -128723, -9583, 259, -1184, 370, -37, 1; ...
%e A121439 Triangle A121412 begins:
%e A121439 1;
%e A121439 1, 1;
%e A121439 3, 1, 1;
%e A121439 18, 4, 1, 1;
%e A121439 170, 30, 5, 1, 1; ...
%e A121439 Row 3 of A121334^-1 equals row 3 of A121412^(-7), which begins:
%e A121439 1;
%e A121439 -7, 1;
%e A121439 7, -7, 1;
%e A121439 -14, 0, -7, 1; ...
%e A121439 Row 4 of A121334^-1 equals row 4 of A121412^(-11), which begins:
%e A121439 1;
%e A121439 -11, 1;
%e A121439 33, -11, 1;
%e A121439 -22, 22, -11, 1;
%e A121439 -143, -22, 11, -11, 1;...
%o A121439 (PARI) /* Matrix Inverse of A121334 */ {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r*(r-1)/2+r-c,r-c)))); return((M^-1)[n+1,k+1])}
%Y A121439 Cf. A121334 (matrix inverse); A121412; variants: A121438, A121440, A121441; A121435 (dual).
%Y A121439 Adjacent sequences: A121436 A121437 A121438 this_sequence A121440 A121441 A121442
%Y A121439 Sequence in context: A119765 A077901 A105619 this_sequence A009205 A086754 A120880
%K A121439 sign,tabl
%O A121439 0,2
%A A121439 Paul D. Hanna (pauldhanna(AT)juno.com), Aug 29 2006
%I A009205
%S A009205 1,1,2,1,2,4,2,1,1,2,2,2,2,4,4,1,2,3,2,6,4,4,2,4,1,2,4,2,2,8,2,3,4,2,4,1,
%T A009205 2,4,4,2,2,8,2,6,6,4,2,2,3,3,4,2,2,8,4,8,4,2,2,12,2,4,2,1,4,8,2,6,4,8,2,
%U A009205 3,2,2,2,2,4,8,2,2,1,2,2,4,4,4,4,4,2,6,4,6,4,4,4,12,2,3,6,1,2,8,2,2,8,2
%N A009205 GCD(d(n), sigma(n)).
%Y A009205 Adjacent sequences: A009202 A009203 A009204 this_sequence A009206 A009207 A009208
%Y A009205 Sequence in context: A077901 A105619 A121439 this_sequence A086754 A120880 A059151
%K A009205 nonn
%O A009205 1,3
%A A009205 David W. Wilson (davidwwilson(AT)comcast.net)
%I A086754
%S A086754 1,1,1,1,1,1,2,1,2,4,2,1,2,1,1,3,3,1,3,9,9,3,3,9,9,3,1,3,3,1,1,4,6,4,1,
%T A086754 4,16,24,16,4,6,24,36,24,6,4,16,24,16,4,1,4,6,4,1,1,5,10,10,5,1,5,25,50,
%U A086754 50,25,5,10,50,100,100,50,10,10,50,100,100,50,10,5,25,50,50,25,5,1,5,10
%N A086754 Pascal's square pyramid read by slices, each slice being read by rows. Each entry in slice n is the sum of the 4 entries above it in slice n-1.
%e A086754 The first 4 slices are
%e A086754 1.11.121.1331
%e A086754 ..11.242.3993
%e A086754 .....121.3993
%e A086754 .........1331
%p A086754 p:=n->seq(seq(binomial(n,i)*binomial(n,j),j=0..n),i=0..n): seq(p(n),n=0..5); (Deutsch)
%o A086754 (PARI) { pt=vector(10,i,matrix(i,i,j,j,1)); for (i=3,10, for (j=2,i-1, pt[i][j,1]=pt[i-1][j-1,1]+pt[i-1][j,1]; pt[i][1,j]=pt[i][j,1]; pt[i][i,j]=pt[i][j,1]; pt[i][j,i]=pt[i][j,1]; ); for(j=2,i-1, for (k=2,i-1, pt[i][j,k]=pt[i-1][j,k]+pt[i-1][j,k-1]+pt[i-1][j-1,k]+pt[i-1][j-1,k-1]))); pt }
%Y A086754 Consider the sequence s[i,j](n) obtained by considering the (i,j)-th entry of the n-th slice. Then if [i,j]= [3,2] we get A006002, if [3,3] we get A000537, if [4,2] we get A004320, if [4,3] we get A004282.
%Y A086754 Cf. A046816.
%Y A086754 Adjacent sequences: A086751 A086752 A086753 this_sequence A086755 A086756 A086757
%Y A086754 Sequence in context: A105619 A121439 A009205 this_sequence A120880 A059151 A059149
%K A086754 nonn,easy
%O A086754 1,7
%A A086754 Jon Perry (perry(AT)globalnet.co.uk), Jul 31 2003
%E A086754 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 18 2004
%I A120880
%S A120880 1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,4,2,1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,4,
%T A120880 2,4,8,4,8,16,8,4,8,4,2,4,2,4,8,4,2,4,2,1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,
%U A120880 4,2,1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,4,2,4,8,4,8,16,8,4,8,4,2,4,2,4,8,4
%N A120880 G.f. satisfies: A(x) = A(x^3)*(1 + 2*x + x^2); thus a(n) = 2^A062756(n), where A062756(n) is the number of 1's in the ternary expansion of n.
%C A120880 More generally, if g.f. of {a(n)} satisfies: A(x) = A(x^3)*(1 + b*x + c*x^2), then a(n) = b^A062756(n)*c^A081603(n), where A062756(n) is the number of 1's, and A081603(n) is the number of 2's, in the ternary expansion of n. This sequence is not the same as A059151.
%C A120880 a(n) is the number of entries in the n-th row of Pascal's triangle that are congruent to 1 mod 3 minus the number of entries that are congruent to 2 mod 3. - Naoki Sato (nsato7(AT)yahoo.ca), Jun 22 2007
%F A120880 a((3^n+1)/2) = 2^n; a(n) = a(floor(n/3))*2^[[n (mod 3)] (mod 2)], with a(0)=1. G.f.: A(x) = prod_{n>=0} (1 + x^(3^n))^2. Self-convolution of A039966. Row sums of triangle A117947(n,k) = balanced ternary of C(n,k) mod 3.
%e A120880 Records are 2^n at positions: 0,1,4,13,40,121,...,(3^n-1)/2,... (n>=0).
%e A120880 A(x) = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 +...
%o A120880 (PARI) a(n)=if(n==0,1,a(n\3)*2^((n%3)%2))
%Y A120880 Cf. A117947, A039966, A062756, A081603.
%Y A120880 Adjacent sequences: A120877 A120878 A120879 this_sequence A120881 A120882 A120883
%Y A120880 Sequence in context: A121439 A009205 A086754 this_sequence A059151 A059149 A013943
%K A120880 nonn
%O A120880 0,2
%A A120880 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 11 2006
%I A059151
%S A059151 1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,4,2,1,2,1,2,4,2,1,2,1,4,8,4,8,16,8,4,
%T A059151 8,4,1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,4,2,1,2,1,2,4,2,1,2,1,8,16,8,16,
%U A059151 32,16,8,16,8,1,2,1,2,4,2,1,2,1,2,4,2,4,8,4,2,4,2,1,2,1,2,4,2,1,2,1,4
%N A059151 A hierarchical sequence (W'3{2,2}* - see A059126).
%H A059151 J. Wallgren, Hierarchical sequences
%Y A059151 Adjacent sequences: A059148 A059149 A059150 this_sequence A059152 A059153 A059154
%Y A059151 Sequence in context: A009205 A086754 A120880 this_sequence A059149 A013943 A082693
%K A059151 easy,nonn
%O A059151 0,2
%A A059151 Jonas Wallgren (jonwa(AT)ida.liu.se), Feb 01 2001
%I A059149
%S A059149 1,2,1,2,4,2,1,2,1,4,8,4,1,2,1,2,4,2,1,2,1,8,16,8,1,2,1,2,4,2,1,2,1,4,
%T A059149 8,4,1,2,1,2,4,2,1,2,1,16,32,16,1,2,1,2,4,2,1,2,1,4,8,4,1,2,1,2,4,2,1,
%U A059149 2,1,8,16,8,1,2,1,2,4,2,1,2,1,4,8,4,1,2,1,2,4,2,1,2,1,32,64,32,1,2,1,2
%N A059149 A hierarchical sequence (W'2{2}* - see A059126).
%H A059149 J. Wallgren, Hierarchical sequences
%Y A059149 Adjacent sequences: A059146 A059147 A059148 this_sequence A059150 A059151 A059152
%Y A059149 Sequence in context: A086754 A120880 A059151 this_sequence A013943 A082693 A097082
%K A059149 easy,nonn
%O A059149 0,2
%A A059149 Jonas Wallgren (jonwa(AT)ida.liu.se), Feb 01 2001
%I A013943
%S A013943 1,2,1,2,4,2,1,2,2,5,4,2,1,2,6,2,6,6,4,2,1,2,4,5,2,8,4,4,4,2,1,2,2,2,3,2,
%T A013943 10,8,6,12,4,2,1,2,6,5,6,4,2,6,7,6,4,11,4,2,1,2,10,2,8,6,8,2,7,5,4,12,6,
%U A013943 4,4,2,1,2,2,5,10,2,6,5,2,8,8,10,16,4,4,11,4,2,1,2,12,2,2,9,6,8,15,2,6,6
%N A013943 Length of period of continued fraction for sqrt(m), m = n-th nonsquare.
%H A013943 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%Y A013943 Cf. A003285, A035015.
%Y A013943 Adjacent sequences: A013940 A013941 A013942 this_sequence A013944 A013945 A013946
%Y A013943 Sequence in context: A120880 A059151 A059149 this_sequence A082693 A097082 A082793
%K A013943 nonn
%O A013943 1,2
%A A013943 Clark Kimberling (ck6(AT)evansville.edu)
%I A082693
%S A082693 1,2,1,2,4,2,1,2,4,8,4,2,1,2,4,8,16,8,4,2,1,2,4,8,16,32,16,8,4,2,1,2,4,
%T A082693 8,16,32,64,32,16,8,4,2,1,2,4,8,16,32,64,128,64,32,16,8,4,2,1,2,4,8,16,
%U A082693 32,64,128,256,128,64,32,16,8,4,2,1,2,4,8,16,32,64,128,256,512,256,128
%N A082693 Pyramidal sequence built with powers of 2.
%Y A082693 Cf. A004738, A082693 (partial sums).
%Y A082693 Adjacent sequences: A082690 A082691 A082692 this_sequence A082694 A082695 A082696
%Y A082693 Sequence in context: A059151 A059149 A013943 this_sequence A097082 A082793 A114929
%K A082693 nonn
%O A082693 1,2
%A A082693 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 12 2003
%I A097082
%S A097082 1,1,1,2,1,2,4,2,1,4,4,20,4,5,1,20,24,8,96,200,24,4,25,1,3,200,48,288,
%T A097082 48,64,2304,1600,10800,288,432,8,675,650,1,26,9,10400,1600,576,2304,
%U A097082 23040,576,2560,1024,368640,516096,128000,2240000,5832000,32256,2304,46656,64,96,91125,3750,84500,6,1,676,9,261
%N A097082 Number of permutations p of (1,2,3,...,n) such that k+p(k) is a Fibonacci number for 1<=k<=n.
%C A097082 See A097083 for the values of n such that a(n)=1.
%Y A097082 Cf. A073364, A095986, A097083.
%Y A097082 Adjacent sequences: A097079 A097080 A097081 this_sequence A097083 A097084 A097085
%Y A097082 Sequence in context: A059149 A013943 A082693 this_sequence A082793 A114929 A144025
%K A097082 nonn
%O A097082 1,4
%A A097082 John W. Layman (layman(AT)math.vt.edu), Jul 23 2004
%E A097082 a(30)-a(50) from Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 26 2004
%E A097082 More terms from David Wasserman (dwasserm(AT)earthlink.net), Dec 19 2007
%I A082793
%S A082793 1,1,1,2,1,2,4,2,2,4,7,4,4,4,7,13,7,8,8,7,13,24,13,14,16,14,13,24,44,24,
%T A082793 26,28,28,26,24,44
%N A082793 A tribonacci triangle in which the top two northeast and southeast diagonals consist of tribonacci numbers.
%C A082793 Uses a Hosoya-like format except that the latter has the Fibonacci recursion. This triangle uses the tribonacci recursion such that every interior number can be obtained by adding the 3 previous numbers, on its diagonal.
%D A082793 Thomas Koshy, <"Fibonacci and Lucas Numbers with Applications">John Wiley and Sons, 2001, Chapter 15, pages 187-195, "Hosoya's Triangle".
%F A082793 T(n, j) = T(n-1, j) + T(n-2, j) + T(n-3, j); (every interior number can be obtained by adding the three previous numbers, on its diagonal.)
%e A082793 T(7,3) = 14 = (8 + 4 + 2) = T(6,3) + T(5,3) + T(4,3).
%Y A082793 Cf. A000073, tribonacci numbers, A058071, Hosoya's triangle.
%Y A082793 Adjacent sequences: A082790 A082791 A082792 this_sequence A082794 A082795 A082796
%Y A082793 Sequence in context: A013943 A082693 A097082 this_sequence A114929 A144025 A058573
%K A082793 nonn
%O A082793 1,4
%A A082793 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 24 2003
%I A114929
%S A114929 0,1,1,2,1,2,4,2,2,4,9,5,1,5,9,21,12,3,3,12,21,51,30,9,1,9,30,51,127,76,
%T A114929 25,4,4,25,76,127,323,196,69,14,1,14,69,196,323,835,512,189,44,5,5,44,
%U A114929 189,512,835,2188,1353,518,133,20,1,20,133,518,1353,2188,5798,3610,1422
%N A114929 Array read by antidiagonals: consider a semi-infinite chessboard with squares labeled (i,j), i >= 0, j >= 0; T(i,j) = number of king-paths of length max{i,j} from (0,0) to (i,j).
%D A114929 Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
%F A114929 Equals Motzkin triangle (A026300) next to same triangle reflected in mirror. See A026300 for the obvious recurrence.
%e A114929 Array begins:
%e A114929 0 1 2 4 9 21 51 ...
%e A114929 1 1 2 5 12 30 ...
%e A114929 2 2 1 3 9 25 ...
%e A114929 4 5 3 1 4 14 ...
%e A114929 ...
%Y A114929 Cf. A026300, A111808, A114972.
%Y A114929 Adjacent sequences: A114926 A114927 A114928 this_sequence A114930 A114931 A114932
%Y A114929 Sequence in context: A082693 A097082 A082793 this_sequence A144025 A058573 A117268
%K A114929 nonn,tabl,easy
%O A114929 0,4
%A A114929 njas, based on May 27 2005 email from Harrie Grondijs (hgrondijs(AT)epo.org), Feb 27 2006
%E A114929 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 20 2006
%I A144025
%S A144025 1,1,1,2,1,2,4,2,2,5,9,4,4,5,13,21,9,8,10,13,35,51,21,18,20,26,35,96,
%T A144025 127,51,42,45,52,70,96,267,323,127,102,105,117,140,192,267,750,835,323,
%U A144025 254,255,273,315,384,534,720,2123,2188,835,646,635,663,735,864,1068
%N A144025 Eigentriangle by rows, A001006(n-k)*A005773(k); 0<=k<=n
%C A144025 Left border = Motzkin numbers, A001006.
%C A144025 Right border = A005773.
%C A144025 Row sums = A005773 shifted: (1, 2, 5, 13, 35, 96, 267,...).
%C A144025 Sum of n-th row terms = rightmost term of next row.
%F A144025 Eigentriangle by rows, A001006(n-k)*A005773(k); 0<=k<=n
%e A144025 First few rows of the triangle =
%e A144025 1;
%e A144025 1, 1;
%e A144025 2, 1, 2;
%e A144025 4, 2, 2, 5;
%e A144025 9, 4, 4, 5, 13;
%e A144025 21, 9, 8, 10, 13, 35;
%e A144025 51, 21, 18, 20, 26, 35, 96;
%e A144025 127, 51, 42, 45, 52, 70, 96, 267;
%e A144025 323, 127, 102, 105, 117, 140, 192, 267, 750;
%e A144025 835, 323, 254, 255, 273, 315, 384, 534, 720, 2123;
%e A144025 ...
%e A144025 Row 3 = (4, 2, 2, 5) = termwise product of (4, 2, 1, 1) and the first 4 terms of A005773: (1, 1, 2, 5) = (4*1, 2*1, 1*2, 1*5). (4, 2, 1, 1) = the first 4 terms of A001066, reversed.
%Y A144025 A001066, Cf. A005773
%Y A144025 Sequence in context: A097082 A082793 A114929 this_sequence A058573 A117268 A119538
%K A144025 nonn,tabl
%O A144025 0,4
%A A144025 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 07 2008
%I A058573
%S A058573 1,0,0,2,1,2,4,2,2,6,4,4,10,6,8,16,9,10,24,14,16,36,20,24,53
%V A058573 1,0,0,2,-1,-2,4,-2,-2,6,-4,-4,10,-6,-8,16,-9,-10,24,-14,-16,36,-20,-24,53
%N A058573 McKay-Thompson series of class 24C for Monster.
%D A058573 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058573 T24C = 1/q + 2*q^2 - q^3 - 2*q^4 + 4*q^5 - 2*q^6 - 2*q^7 + 6*q^8 - 4*q^9 - ...
%Y A058573 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058573 Adjacent sequences: A058570 A058571 A058572 this_sequence A058574 A058575 A058576
%Y A058573 Sequence in context: A082793 A114929 A144025 this_sequence A117268 A119538 A068309
%K A058573 sign
%O A058573 -1,4
%A A058573 njas, Nov 27, 2000
%I A117268
%S A117268 1,1,1,2,1,2,4,2,3,5,7,4,6,9,14,13,7,11,17,26,40,24,13,20,31,48,74,114,
%T A117268 44,24,37,57,88,136,210,324,81,44,68,105,162,250,386,596,920
%N A117268 Triangle, row sums = A117289, binomial transform of the tribonacci sequence.
%C A117268 Leftmost column of the triangle = the tribonacci sequence, A000073: (1, 1, 2, 4, 7, 13...); rightmost diagonal of the triangle = binomial transform of A000073 = (1, 2, 5, 14, 40, 114...) = A117189.
%F A117268 Difference rows of A117267 become rows of A117268
%e A117268 Taking difference rows of A117267: (1; 1, 2; 2, 3, 5; 4, 6, 9, 14;...), we get A117268:
%e A117268 1;
%e A117268 1, 1;
%e A117268 2, 1, 2;
%e A117268 4, 2, 3, 5;
%e A117268 7, 4, 6, 9, 14;
%e A117268 13, 7, 11, 17, 26, 40;
%e A117268 24, 13, 20, 31, 48, 74, 114;
%e A117268 ...
%Y A117268 Cf. A117267, A117189, A000073.
%Y A117268 Adjacent sequences: A117265 A117266 A117267 this_sequence A117269 A117270 A117271
%Y A117268 Sequence in context: A114929 A144025 A058573 this_sequence A119538 A068309 A099470
%K A117268 nonn
%O A117268 1,4
%A A117268 Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 05 2006
%I A119538
%S A119538 0,1,2,1,2,4,2,3,6,1,2,4,2,3,6,4,5,10,2,3,6,3,4,8,6,7,14,1,2,4,2,3,6,4,
%T A119538 5,10,2,3,6,3,4,8,6,7,14,4,5,10,5,6,12,10,11,22,2,3,6,3,4,8,6,7,14,3,4,
%U A119538 8,4,5,10,8,9,18,6,7,14,7,8,16,14,15,30,1,2,4,2,3,6,4,5,10,2,3,6,3,4,8
%N A119538 Fixed point of the morphism a -> {a, a + 1, 2a + 2} beginning with 0.
%t A119538 Nest[Flatten[ #1 /. a_Integer -> {a, a + 1, 2a + 2}] &, {0}, 5]
%Y A119538 Cf. A000120.
%Y A119538 Adjacent sequences: A119535 A119536 A119537 this_sequence A119539 A119540 A119541
%Y A119538 Sequence in context: A144025 A058573 A117268 this_sequence A068309 A099470 A099254
%K A119538 nonn
%O A119538 1,3
%A A119538 Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 27 2006
%I A068309
%S A068309 1,2,1,2,4,2,3,6,3,2,6,4,3,6,3,4,8,4,6,8,3,6,10,2,7,10,4,6,10,2,5,10,5,
%T A068309 6,10
%N A068309 Number of different lucky numbers occurring when 2n is expressed as p1+q1 = ... = pk+qk where pk,qk are lucky numbers with pk <= qk.
%C A068309 n=5: 10 = 1+9 = 3+7, we can see 4 different lucky numbers. so a(5) = 4.
%Y A068309 Cf. A000959, A002850.
%Y A068309 Adjacent sequences: A068306 A068307 A068308 this_sequence A068310 A068311 A068312
%Y A068309 Sequence in context: A058573 A117268 A119538 this_sequence A099470 A099254 A121339
%K A068309 easy,nonn
%O A068309 1,2
%A A068309 Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 25 2002
%I A099470
%S A099470 1,2,1,2,4,2,3,6,3,4,8,4,5,10,5,6,12,6,7,14,7,8,16,8,9,18,9,10,20,10,11,
%T A099470 22,11,12,24,12,13,26,13,14,28,14,15,30,15,16,32,16,17,34,17,18,36,18
%V A099470 -1,-2,-1,2,4,2,-3,-6,-3,4,8,4,-5,-10,-5,6,12,6,-7,-14,-7,8,16,8,-9,-18,-9,10,20,10,
%W A099470 -11,-22,-11,12,24,12,-13,-26,-13,14,28,14,-15,-30,-15,16,32,16,-17,-34,-17,18,36,18
%N A099470 A sequence generated from the Quadrifoil.
%C A099470 a(3n - 1) = 2n (unsigned; n = 1, 2, 3...). In A099471, a(3n) = (2n + 1), unsigned. Odifreddi, p. 135: "Since the trefoil has polynomial x^2 - x + 1 and the quadrifoil (or flat knot) is the sum of two trefoils, its polynomial is (x^2 - x + 1)^2 = x^4 - 2x^3 + 3x^2 - 2x + 1."
%D A099470 Piergiorgio Odifreddi: "The Mathematical Century; The 30 Greatest Problems of the Last 100 Years"; Princeton University Press, 2000; p. 135.
%F A099470 M = the 4 X 4 companion matrix to the Quadrafoil polynomial x^4 - 2x^3 + 3x^2 - 2x + 1: [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 2 -3 2]. a(n) = rightmost term in M^n * [1 0 0 0].
%F A099470 O.g.f.: -x/(x^2-x+1)^2. a(n) = 2a(n-1)-3a(n-2)+2a(n-3)-a(n-4) = -A099254(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 06 2008
%e A099470 a(7) = -3 since M^7 * [1 0 0 0] = [2 4 2 -3].
%Y A099470 Cf. A099471.
%Y A099470 Adjacent sequences: A099467 A099468 A099469 this_sequence A099471 A099472 A099473
%Y A099470 Sequence in context: A117268 A119538 A068309 this_sequence A099254 A121339 A099500
%K A099470 sign
%O A099470 1,2
%A A099470 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2004
%I A099254
%S A099254 1,2,1,2,4,2,3,6,3,4,8,4,5,10,5,6,12,6,7,14,7,8,16,8,9,18,9,10,20,10,11,
%T A099254 22,11,12,24,12,13,26,13,14,28,14,15,30,15,16,32,16,17,34,17,18,36,18,
%U A099254 19,38,19,20,40,20,21,42,21
%V A099254 1,2,1,-2,-4,-2,3,6,3,-4,-8,-4,5,10,5,-6,-12,-6,7,14,7,-8,-16,-8,9,18,9,-10,-20,-10,11,
%W A099254 22,11,-12,-24,-12,13,26,13,-14,-28,-14,15,30,15,-16,-32,-16,17,34,17,-18,-36,-18,19,
%X A099254 38,19,-20,-40,-20,21,42,21
%N A099254 Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot).
%C A099254 A granny knot sequence.
%F A099254 G.f.: 1/(1-2x+3x^2-2x^3+x^4)=1/(1-x+x^2)^2; a(n)=4sqrt(3)*sin(pi*n/3+pi/3)/9+2(n + 1)sin(pi*n/3+pi/6)/3.
%F A099254 a(n)=sum{k=0..floor(n/2), binomial(n-k,k)*(n-k+1)*(-1)^k}. - Paul Barry (pbarry(AT)wit.ie), Nov 12 2004
%Y A099254 Row sums of array A128502.
%Y A099254 Adjacent sequences: A099251 A099252 A099253 this_sequence A099255 A099256 A099257
%Y A099254 Sequence in context: A119538 A068309 A099470 this_sequence A121339 A099500 A120253
%K A099254 easy,sign
%O A099254 0,2
%A A099254 Paul Barry (pbarry(AT)wit.ie), Oct 08 2004
%I A121339
%S A121339 2,1,2,4,2,4,1,1,1,4,1,4,6,3,6,2,6,1,1,1,1,6,1,2,1,6,1,6,8,4,8,2,1,3,1,
%T A121339 2,8,2,8,1,1,2,1,1,8,1,2,4,2,1,8,1,3,1,8,1,8,10,5,10,3,2,3,10,2,1,1,2,
%U A121339 10,2,10,1,1,3,5,3,1,1,10,1,1,1,10,1,2,1,10,1,4,1,10,1,10,12,6,12,4,12
%N A121339 Periodic part of continued fraction for square roots of integers.
%H A121339 T. D. Noe, Rows n=2..1000 of triangle, flattened
%e A121339 The table starts:
%e A121339 2
%e A121339 1 2
%e A121339
%e A121339 4
%e A121339 2 4
%e A121339 1 1 1 4
%e A121339 1 4
%Y A121339 Cf. A003285 (row lengths), A013943 (row lengths for non-empty rows).
%Y A121339 Adjacent sequences: A121336 A121337 A121338 this_sequence A121340 A121341 A121342
%Y A121339 Sequence in context: A068309 A099470 A099254 this_sequence A099500 A120253 A060547
%K A121339 nice,nonn,tabf
%O A121339 2,1
%A A121339 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006
%I A099500
%S A099500 1,1,1,1,2,1,2,4,2,4,4,4,8,4,32,8,16,16,32,16,32,128,32,64,59,124,499,
%T A099500 32,478,122,248,1886,239,1980,240,944,500,6181,3848,970,3776,3844,6167,
%U A099500 13774,3806,27985,24562,4016,102193,13224,105671,12565,399445,29436
%N A099500 Number of distinct products of subsets of integers in the interval [n^2+1, (n+1)^2-1] which are twice a square.
%C A099500 Granville and Selfridge prove Kaplansky's conjecture that a(n) > 0 for all n. Interestingly, the first 24 terms are powers of 2, but the 25th term is not.
%C A099500 a(n) <= A120253(n) = 2^A120254(n). The sequences are equal iff each subset has a distinct product. - Martin Fuller (martin_n_fuller(AT)btinternet.com), Jun 13 2006
%H A099500 Andrew Granville and John Selfridge, Product of integers in an interval, modulo squares (pdf), Electronic Journal of Combinatorics, Volume 8(1), 2001.
%e A099500 a(5) = 2 because the interval [26,35] contains two sets of such integers: {32} and {27,28,30,35}.
%e A099500 a(25) < A120253(25) because there are distinct subsets with the same product. The smallest is 627*630*646*648*650*660*663*672 = 627*637*640*646*648*660*663*675 = 2*125707982400^2.
%t A099500 Needs["DiscreteMath`Combinatorica`"]; Table[lst=Range[n^2+1, (n+1)^2-1]; x=Times@@lst; {p, e}=Transpose[FactorInteger[x]]; Do[If[e[[i]]==1 && p[[i]]!=2, lst=DeleteCases[lst, _?(Mod[ #, p[[i]]]==0&)]], {i, Length[p]}]; i=1; cnt=0; xLst={}; While[i<2^Length[lst], ss=NthSubset[i, lst]; x=Times@@ss; If[Mod[x, 2]==0 && IntegerQ[Sqrt[x/2]] && !MemberQ[xLst, x], cnt++; AppendTo[xLst, x] (*; Print[{n, ss}]*)]; i++ ]; cnt, {n, 31}]
%Y A099500 Cf. A099501 (size of the subset having the least number of integers).
%Y A099500 Cf. A120253 number of subsets whose product is twice a square.
%Y A099500 Adjacent sequences: A099497 A099498 A099499 this_sequence A099501 A099502 A099503
%Y A099500 Sequence in context: A099470 A099254 A121339 this_sequence A120253 A060547 A079878
%K A099500 nonn
%O A099500 1,5
%A A099500 T. D. Noe (noe(AT)sspectra.com), Oct 20 2004
%E A099500 More terms from Martin Fuller (martin_n_fuller(AT)btinternet.com), Jun 13 2006
%I A120253
%S A120253 1,1,1,1,2,1,2,4,2,4,4,4,8,4,32,8,16,16,32,16,32,128,32,64,64,128,512,
%T A120253 32,512,128,256,2048,256,2048,256,1024,512,8192,4096,1024,4096,4096,
%U A120253 8192,16384,4096,32768,32768,4096,131072,16384,131072,16384,524288
%N A120253 Number of subsets of integers in the interval [n^2+1, (n+1)^2-1] whose product is twice a square.
%C A120253 Also the number of subsets in the same interval whose product is precisely a square, if 1 is included.
%H A120253 Andrew Granville and John Selfridge, Product of integers in an interval, modulo squares (pdf), Electronic Journal of Combinatorics, Volume 8(1), 2001.
%F A120253 2^A120254(n)
%e A120253 a(5) = 2 because the interval [26,35] contains two sets of such integers: {32} and {27,28,30,35}.
%Y A120253 Cf. A120254. A099500 is the number of distinct products which are twice a square. A099501 is the smallest size of a subset which is twice a square.
%Y A120253 Adjacent sequences: A120250 A120251 A120252 this_sequence A120254 A120255 A120256
%Y A120253 Sequence in context: A099254 A121339 A099500 this_sequence A060547 A079878 A137406
%K A120253 nonn
%O A120253 1,5
%A A120253 Martin Fuller (martin_n_fuller(AT)btinternet.com), Jun 13 2006
%I A060547
%S A060547 2,1,2,4,2,4,8,4,8,16,8,16,32,16,32,64,32,64,128,64,128,256,128,256,
%T A060547 512,256,512,1024,512,1024,2048,1024,2048,4096,2048,4096,8192,4096,
%U A060547 8192,16384,8192,16384,32768,16384,32768,65536,32768,65536,131072
%N A060547 a(n) is the number of patterns, invariant under 120 degree rotations, that may appear in a top-down equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement.
%C A060547 The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
%D A060547 A. Barb\'{e}, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
%H A060547 Index entries for sequences related to cellular automata
%F A060547 a(n) = 2^[floor(n/3)+(n mod 3)mod 2]
%p A060547 gf := (1+x^2+x^4)/(1-x^3)^2: s := series(gf, x, 100): for i from 0 to 70 do printf(`%d,`,2^coeff(s,x,i)) od:
%Y A060547 a(n)=2^A008611(n-1) for n >= 1. Cf. A060550.
%Y A060547 Adjacent sequences: A060544 A060545 A060546 this_sequence A060548 A060549 A060550
%Y A060547 Sequence in context: A121339 A099500 A120253 this_sequence A079878 A137406 A120855
%K A060547 easy,nice,nonn
%O A060547 1,1
%A A060547 Andr\'{e} Barb\'{e} (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
%E A060547 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 04 2001
%I A079878
%S A079878 1,2,1,2,4,2,4,8,7,4,8,4,8,2,4,8,16,14,9,18,15,8,16,8,16,6,12,24,19,8,
%T A079878 16,32,31,28,21,6,12,24,9,18,36,30,17,34,23,46,45,42,35,20,40,28,3,6,12,
%U A079878 24,48,38,17,34,7,14,28,56,47,28,56,44,19,38,5,10,20,40,5,10,20,40,1,2
%N A079878 a(1)=1, then a(n)=2*a(n-1) if 2*a(n-1)<=n, a(n)=2*a(n-1)-n otherwise.
%C A079878 For n=1, 2, 8, 32, 46, 392, 12230, .... we have a(n)=n. For n=1, 3, 79, 235, 431, 1503, 2943, 6059, 6619, ... we have a(n)=1. Memo: these should be submitted as separate sequences. - njas
%F A079878 It seems that sum(k=1, n, a(k))/n^2 ->1/4
%o A079878 (PARI) a=1; for(n=2,100,b=if(sign(2*a-n)-1,2*a,2*a-n); a=b; print1(b,","))
%Y A079878 A064434(n) + 1.
%Y A079878 Adjacent sequences: A079875 A079876 A079877 this_sequence A079879 A079880 A079881
%Y A079878 Sequence in context: A099500 A120253 A060547 this_sequence A137406 A120855 A091173
%K A079878 nonn
%O A079878 1,2
%A A079878 Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 20 2003
%I A137406
%S A137406 1,1,2,1,2,4,2,6,8,8,1,6,16,16,16,3,14,36,56,48,32,1,12,44,88,128,96,64,
%T A137406 4,28,104,232,352,384,256,128,1,20,100,296,592,800,832,512,256,5,50,244,728,
%U A137406 1536,2368,2688,2304,1280,512,1,30,200,784,2048,3872,5568,5888,4864,2560,1024
%V A137406 1,1,-2,-1,2,-4,-2,6,-8,8,1,-6,16,-16,16,3,-14,36,-56,48,-32,-1,12,-44,88,-128,96,-64,
%W A137406 -4,28,-104,232,-352,384,-256,128,1,-20,100,-296,592,-800,832,-512,256,5,-50,244,-728,
%X A137406 1536,-2368,2688,-2304,1280,-512,-1,30,-200,784,-2048,3872,-5568,5888,-4864,2560,-1024
%N A137406 Triangular sequence from coefficients of a switched even -odd polynomial recursion: Even:p(x,n)=2*x*p(x, n - 1) - p(x, n - 2); Odd:p(x,n)=(1 - 2*x)*p(x, n - 1) - p(x, n - 2);.
%C A137406 A002530 is the row sums:
%C A137406 {1, -1, -3, 4, 11, -15, -41, 56, 153, -209, -571}
%F A137406 p(x,-1)=0;p(x,0)=1;p(x,1]=1-28x; Even:p(x,n)=2*x*p(x, n - 1) - p(x, n - 2); Odd:p(x,n)=(1 - 2*x)*p(x, n - 1) - p(x, n - 2);
%e A137406 {1},
%e A137406 {1, -2},
%e A137406 {-1, 2, -4},
%e A137406 {-2, 6, -8, 8},
%e A137406 {1, -6, 16, -16, 16},
%e A137406 {3, -14, 36, -56, 48, -32},
%e A137406 {-1, 12, -44, 88, -128, 96, -64},
%e A137406 {-4, 28, -104, 232, -352, 384, -256, 128},
%e A137406 {1, -20, 100, -296, 592, -800, 832, -512, 256},
%e A137406 {5, -50, 244, -728, 1536, -2368, 2688, -2304, 1280, -512},
%e A137406 {-1, 30, -200, 784, -2048, 3872, -5568, 5888, -4864, 2560, -1024}
%t A137406 Clear[p, x, a] p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 1 - 2*x; p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, 2*x*p[x, n - 1] - p[x, n - 2], (1 - 2*x)*p[x, n - 1] - p[x, n - 2]]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]
%Y A137406 Cf. A002530.
%Y A137406 Adjacent sequences: A137403 A137404 A137405 this_sequence A137407 A137408 A137409
%Y A137406 Sequence in context: A120253 A060547 A079878 this_sequence A120855 A091173 A101897
%K A137406 tabl,uned,sign
%O A137406 1,3
%A A137406 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 14 2008
%I A120855
%S A120855 0,2,1,2,4,3,1,3,2,2,4,3,4,6,5,3,5,4,1,3,2,3,5,4,2,4,3,2,4,3,4,6,5,3,5,
%T A120855 4,4,6,5,6,8,7,5,7,6,3,5,4,5,7,6,4,6,5,1,3,2,3,5,4,2,4,3,3,5,4,5,7,6,4,
%U A120855 6,5,2,4,3,4,6,5,3,5,4,2,4,3,4,6,5,3,5,4,4,6,5,6,8,7,5,7,6,3,5,4,5,7,6
%N A120855 Row sums of triangle A120854, which is the matrix log of triangle A117939.
%C A120855 Triangle A117939 is related to powers of 3 partitions of n, and is the matrix square of A117947(n,k) = balanced ternary digits of C(n,k) mod 3, also A117947(n,k) = L(C(n,k)/3) where L(j/p) is the Legendre symbol of j and p.
%F A120855 a(n) = 2*A062756 + A081603(n), where A062756(n) = number of 1's in ternary expansion of n, and A081603(n) = number of 2's in ternary expansion of n.
%o A120855 (PARI) {a(n)=local(M=matrix(n+1,n+1,r,c,(binomial(r-1,c-1)+1)%3-1)^2, L=sum(i=1,#M,-(M^0-M)^i/i));return(sum(k=0,n,L[n+1,k+1]))}
%Y A120855 Cf. A120854, A117947; A062756, A081603, A053735.
%Y A120855 Adjacent sequences: A120852 A120853 A120854 this_sequence A120856 A120857 A120858
%Y A120855 Sequence in context: A060547 A079878 A137406 this_sequence A091173 A101897 A078142
%K A120855 nonn
%O A120855 0,2
%A A120855 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 08 2006
%I A091173
%S A091173 1,1,1,1,2,1,2,4,3,1,4,10,9,4,1,10,28,30,16,5,1,30,90,108,68,25,6,1,106,
%T A091173 328,426,304,130,36,7,1,420,1338,1842,1444,700,222,49,8,1,1818,6024,
%U A091173 8706,7320,3930,1404,350,64,9,1,8530,29626,44736,39700,23110,9150,2548
%N A091173 Triangle, read by rows, where the n-th row lists the coefficients of the polynomial of degree n, with root -1, that generates the n-th diagonal of this sequence.
%C A091173 The left-most column (A091174) is determined by the condition that the root of each row polynomial is -1. The next column is T(n,1)=A091175(n+1) (n>=0).
%F A091173 T(n+k, k) = Sum T(n, j)*k^j {j=0..n}, with T(0, 0)=1, T(0, n)=1, and T(n, 0) = -sum T(n, j)*(-1)^j {j=1..n}.
%e A091173 For row n=3, k=2, T(n+k,k)=T(5,2)=30=(2)+(4)2+(3)2^2+(1)2^3.
%e A091173 For n=4, k=3, T(n+k,k)=T(7,3)=304=(4)+(10)3+(9)3^2+(4)3^3+(1)3^4.
%e A091173 Rows begin with n=0:
%e A091173 {1},
%e A091173 {1,1},
%e A091173 {1,2,1},
%e A091173 {2,4,3,1},
%e A091173 {4,10,9,4,1},
%e A091173 {10,28,30,16,5,1},
%e A091173 {30,90,108,68,25,6,1},
%e A091173 {106,328,426,304,130,36,7,1},
%e A091173 {420,1338,1842,1444,700,222,49,8,1},
%e A091173 {1818,6024,8706,7320,3930,1404,350,64,9,1},...
%Y A091173 Cf. A091174, A091175.
%Y A091173 Adjacent sequences: A091170 A091171 A091172 this_sequence A091174 A091175 A091176
%Y A091173 Sequence in context: A079878 A137406 A120855 this_sequence A101897 A078142 A133422
%K A091173 nonn,tabl
%O A091173 0,5
%A A091173 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 25 2003
%I A101897
%S A101897 1,1,1,1,2,1,2,4,3,1,5,11,9,4,1,17,38,33,16,5,1,71,162,145,74,25,6,1,357,
%T A101897 824,753,396,140,36,7,1,2101,4892,4535,2434,885,237,49,8,1,14203,33286,31185,
%U A101897 16982,6295,1730,371,64,9,1,108609,255824,241621,133012,50001,13992,3073,548,81
%V A101897 1,-1,1,1,-2,1,-2,4,-3,1,5,-11,9,-4,1,-17,38,-33,16,-5,1,71,-162,145,-74,25,-6,1,-357,
%W A101897 824,-753,396,-140,36,-7,1,2101,-4892,4535,-2434,885,-237,49,-8,1,-14203,33286,-31185,
%X A101897 16982,-6295,1730,-371,64,-9,1,108609,-255824,241621,-133012,50001,-13992,3073,-548,81
%N A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0, and where T^k is the k-th power of T as a lower triangular matrix.
%C A101897 Column 0 forms A101900. Absolute row sums form A101901.
%F A101897 T(n, k) = Sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(0, 0)=1 and T(n, 0) = -Sum_{j=1, n} T(n, j) for n>0.
%e A101897 Rows begin:
%e A101897 [1],
%e A101897 [ -1,1],
%e A101897 [1,-2,1],
%e A101897 [ -2,4,-3,1],
%e A101897 [5,-11,9,-4,1],
%e A101897 [ -17,38,-33,16,-5,1],
%e A101897 [71,-162,145,-74,25,-6,1],
%e A101897 [ -357,824,-753,396,-140,36,-7,1],
%e A101897 [2101,-4892,4535,-2434,885,-237,49,-8,1],
%e A101897 [ -14203,33286,-31185,16982,-6295,1730,-371,64,-9,1],...
%o A101897 (PARI) {T(n,k)=if(k>n|n<0|k<0,0,if(k==n,1, if(k==0,-sum(j=1,n,T(n,j)), sum(j=0,n-k,T(n-k,j)*T(j+k-1,k-1));));)}
%Y A101897 Cf. A091351, A101900, A101901, A101898, A101899.
%Y A101897 Adjacent sequences: A101894 A101895 A101896 this_sequence A101898 A101899 A101900
%Y A101897 Sequence in context: A137406 A120855 A091173 this_sequence A078142 A133422 A099312
%K A101897 sign,tabl
%O A101897 0,5
%A A101897 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 20 2004
%I A078142
%S A078142 0,2,1,2,4,3,2,2,1,6,5,3,3,4,5,2,8,3,6,6,3,7,2,3,4,5,1,4,7,7,5,2,6,10,
%T A078142 6,3,12,8,4,6,8,5,6,7,5,4,2,3,2,6,9,5,11,3,9,4,7,9,5,7,3,7,3,2,7,8,14,
%U A078142 10,3,8,10,3,8,14,5,8,7,6,2,6,1,10,17,5,12,8,8,7,11,7,5,4,6,4,10,3,3,4
%N A078142 a(n) = the sum of the differences of the distinct prime factors p of n and the next square larger than p.
%e A078142 a(6)=3 because 6=2*3 and (4-2)+(4-3)=3. a(20)=6 because 20=2^2*5 and (4-2)+(9-5)=6.
%Y A078142 Adjacent sequences: A078139 A078140 A078141 this_sequence A078143 A078144 A078145
%Y A078142 Sequence in context: A120855 A091173 A101897 this_sequence A133422 A099312 A117505
%K A078142 easy,nonn
%O A078142 1,2
%A A078142 Jason Earls (zevi_35711(AT)yahoo.com), Nov 20 2002
%I A133422
%S A133422 0,1,1,2,1,2,4,3,2,2,12,3,9,5,2,4,6,3,21,3,5,13,6,4,2,10,3,6,10,3,47,5,
%T A133422 13,7,5,4,13,22,10,4,8,6,46,14,3,7,16,5,50,3
%N A133422 Number of steps to reach 1 under repeated applications of the "7x+1" map of A133421, or -1 if 1 is never reached.
%C A133422 The 7x+1 map sends x to x/2 if x is even, x/3 if x is divisible by 3, x/5 if x is divisible by 5, otherwise 7x+1.
%H A133422 Tomas Oliveira e Silva, The px+1 problem
%H A133422 Index entries for sequences related to 3x+1 (or Collatz) problem
%t A133422 nxt[x_]:= Which[Mod[x,2]==0,x/2,Mod[x,3]==0,x/3,Mod[x,5]==0,x/5,True,7x+1]; Table[First[First[Position[NestList[nxt,i,100],1]]]-1,{i,50}] - Harvey P. Dale (hpd1(AT)nyu.edu), Dec 27 2007
%Y A133422 Cf. A133421, A133419, ...
%Y A133422 Adjacent sequences: A133419 A133420 A133421 this_sequence A133423 A133424 A133425
%Y A133422 Sequence in context: A091173 A101897 A078142 this_sequence A099312 A117505 A131380
%K A133422 nonn,easy,more
%O A133422 1,4
%A A133422 njas, Nov 27 2007
%E A133422 More terms from Harvey P. Dale (hpd1(AT)nyu.edu), Dec 27 2007
%I A099312
%S A099312 0,1,2,1,2,4,3,2,4,2,6,3,4,2,6,4,3,6,4,3,2,6,10,2,4,4,3,6,10,4,12,4,6,2,
%T A099312 4,6,3,10,5,12,6,4,10,6,4,6,3,4,12,2,4,10,6,5,4,6,12,16,10,3,6,10,5,6,4,
%U A099312 4,6,12,16,6,4,10,6,18,4,10,12,5,5,10,12,4,5,16
%N A099312 Exponent of greatest power of 2 dividing the n-th minimal number.
%C A099312 A minimal number is the smallest number with a given number of divisors, see A007416;
%C A099312 a(n) = A007814(A007416(n)).
%Y A099312 Cf. A099311, A099314, A099316.
%Y A099312 Adjacent sequences: A099309 A099310 A099311 this_sequence A099313 A099314 A099315
%Y A099312 Sequence in context: A101897 A078142 A133422 this_sequence A117505 A131380 A100461
%K A099312 nonn
%O A099312 1,3
%A A099312 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 12 2004
%I A117505
%S A117505 1,2,1,2,4,3,2,4,16,13,2,4,16,80,67,2,4,16,80,448,381,2,4,16,80,448,
%T A117505 2688,2307,2,4,16,80,448,2688,16896,14589,2,4,16,80,448,2688,16896,
%U A117505 109824,95235,2,4,16,80,448,2688
%N A117505 Triangle of coefficients for polynomials used for the column g.f.s of triangle A116880, called CM(1,2).
%C A117505 The g.f. G(m,x) for column m=1,2,... of triangle A116880=CM(1,2) is x*(-sum(a(m,k)*x^(k-1),k=1..m) + sum(a(m,k)*x^k,k=0..m)*2*c(2*x))/(1+x), with the o.g.f. c(x) of A000108 (Catalan numbers).
%H A117505 W. Lang: First 10 rows.
%F A117505 a(m,m)= A064062(m) =:C(2;m), m>=0, and a(m,k)=2*A052701(k) = C(k)*2^(k+1), for k=1,...,m-1, and C(k):=A000108(k) (Catalan).
%e A117505 m=3: G(3,x)= x*(-(4+16*x+13*x^2) +
%e A117505 (2+4*x+16*x^2+13*x^3)*2*c(2*x))/(1+x).
%Y A117505 Adjacent sequences: A117502 A117503 A117504 this_sequence A117506 A117507 A117508
%Y A117505 Sequence in context: A078142 A133422 A099312 this_sequence A131380 A100461 A124904
%K A117505 nonn,easy,tabl
%O A117505 0,2
%A A117505 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 13 2006
%I A131380
%S A131380 0,2,1,2,4,3,4,6,5,6,8,7,8,10,9,10,12,11,12,14,13,14,16,15,16,18,17,18,
%T A131380 20,19,20,22,21,22,24,23,24,26,25,26,28,27,28,30,29,30,32,31,32,34,33,
%U A131380 34,36,35,36,38,37,38,40,39,40,42,41,42,44,43,44,46,45,46,48,47,48,50
%N A131380 a(3n)=2n, a(3n+1)=2n+2, a(3n+2)=2n+1.
%Y A131380 Adjacent sequences: A131377 A131378 A131379 this_sequence A131381 A131382 A131383
%Y A131380 Sequence in context: A133422 A099312 A117505 this_sequence A100461 A124904 A129144
%K A131380 nonn
%O A131380 0,2
%A A131380 Paul Curtz (bpcrtz(AT)free.fr), Oct 01 2007
%I A100461
%S A100461 1,1,2,1,2,4,3,4,6,8,7,8,9,12,16,25,26,27,28,30,32,49,50,51,52,55,
%T A100461 60,64,109,110,111,112,115,120,126,128,229,230,231,232,235,240,245,
%U A100461 248,256,481,482,483,484,485,486,490,496,504,512,1003,1004,1005
%N A100461 Triangle read by rows, based on array described below.
%C A100461 Form an array a(m,n) (n >= 1, 1 <= m <= n) by: a(1,n) = 2^(n-1) for all n; a(m+1,n) = (n-m)*floor( (a(m,n)-1)/(n-m) ) for 1 <= m <= n-1.
%e A100461 Array begins:
%e A100461 1 2 4 8 16 32...
%e A100461 * 1 2 6 12 30...
%e A100461 * * 1 4 9 28...
%e A100461 * * * 3 8 27...
%e A100461 * * * * 7 26...
%e A100461 * * * * * 25...
%e A100461 and triangle begins:
%e A100461 1
%e A100461 1 2
%e A100461 1 2 4
%e A100461 3 4 6 8
%e A100461 7 8 9 12 16
%e A100461 ...
%Y A100461 Cf. A100452, A100462, A119444.
%Y A100461 Adjacent sequences: A100458 A100459 A100460 this_sequence A100462 A100463 A100464
%Y A100461 Sequence in context: A099312 A117505 A131380 this_sequence A124904 A129144 A105022
%K A100461 nonn,tabl
%O A100461 1,3
%A A100461 njas, Nov 22 2004
%I A124904
%S A124904 0,2,1,2,4,3,5,7,2,4,6,8,3,10,5,12,7,14,9,4,11,6,18,13,8,32,3,10,22,5,
%T A124904 29,12,36,7,31,14,38,9,33,4,16,40,11,35,6,18,42,13,25,49,8,32,44,15,27,
%U A124904 39,10,22,34,5,17,29,41,12,24,36,48,7,19,31,43,14,26,38,91,9,21,33,86
%N A124904 a(n) = least integer j>=0 such that n = floor[(3^j)/(2^k)] for some integer k>=0.
%C A124904 The k-sequence is A124912.
%e A124904 1=[3^0/2^0], 2=[3^2/2^2], 3=[3^1/2^0], 4=[3^2/2^1],...,
%e A124904 so j-sequence = (0,2,1,2,...); k-sequence = (0,2,0,1,...).
%Y A124904 Cf. A124912.
%Y A124904 Adjacent sequences: A124901 A124902 A124903 this_sequence A124905 A124906 A124907
%Y A124904 Sequence in context: A117505 A131380 A100461 this_sequence A129144 A105022 A105970
%K A124904 nonn
%O A124904 1,2
%A A124904 Clark Kimberling (ck6(AT)evansville.edu), Nov 12 2006
%I A129144
%S A129144 1,1,1,2,1,2,4,3,6,5,1,2,4,8,5,10,7,14,13,9,18,17,13,3,6,12,24,23,19,9,
%T A129144 18,5,10,20,7,14,28,21,5,10,20,1,2,4,8,16,32,23,3,6,12,24,3,6,12,24,1,2,
%U A129144 4,8,16,32,15,30,9,18,36,19,38,21,42,27,54,49,37,11,22,44,23,46,25,50
%N A129144 a(1)=a(2)=1; for n>2: a(n) = (s = sum of all previous terms) minus (maximal square less than s).
%C A129144 Positions of first 20 numbers in the sequence: {1:{1, 2, 3, 5, 11, 42, 57, 141, 709, 1119, 1223, 1333}}, {2:{4, 6, 12, 43, 58, 142, 710, 1120, 1224, 1334}}, {3:{8, 24, 49, 53, 683, 1550}}, {4:{7, 13, 44, 59, 143, 711, 1121, 1225, 1335}}, {5:{10, 15, 32, 39, 220, 361, 402, 562, 672}}, {6:{9, 25, 50, 54, 684, 1551}}, {7:{17, 35, 99, 188, 353, 1156, 1630}}, {8:{14, 45, 60, 144, 712, 1122, 1226, 1336}}, {9:{20, 30, 65, 164, 236, 723}}, {10:{16, 33, 40, 221, 362, 403, 563, 673}}, {11:{76, 277, 333}}, {12:{26, 51, 55, 685, 1552}}, {13:{19, 23, 318, 338}}, {14:{18, 36, 100, 189, 354, 1157, 1631}}, {15:{63, 430, 613, 1541, 1754}}, {16:{46, 61, 145, 713, 1123, 1227, 1337}}, {17:{22, 107, 158, 325, 1914}}, {18:{21, 31, 66, 165, 237, 724}}, {19:{29, 68, 103}}, {20:{34, 41, 222, 363, 404, 564, 674}}. From the first 2000 terms, the first absent numbers are: 67,105,109,134,148,152,168,183,187,193,199. Do all numbers, except zero, appear eventually in the sequence? Do ones appear infinitely many times?
%H A129144 Zak Seidov, Table of n, a(n) for n = 1..2000
%F A129144 a(1)=a(2)=1; for n>2: a(n)=s-(floor(sqrt(s)))^2, where s=sum (a(i), i=1..n-1).
%t A129144 a={1,1};s=2;Do[b=s-(Floor[Sqrt[s]])^2;AppendTo[a,b];s=s+b,{2000}];a
%Y A129144 Adjacent sequences: A129141 A129142 A129143 this_sequence A129145 A129146 A129147
%Y A129144 Sequence in context: A131380 A100461 A124904 this_sequence A105022 A105970 A133950
%K A129144 nonn
%O A129144 1,4
%A A129144 Zak Seidov (zakseidov(AT)gmail.com), Mar 31 2007
%I A105022
%S A105022 0,0,1,2,1,2,4,4,1,2,4,8,4,8,8,8,1,2,4,8,4,8,12,16,4,8,12,16,8,16,16,16,
%T A105022 1,2,4,8,4,8,12,16,4,8,12,24,12,24,24,32,4,8,12,16,12,24,24,32,8,16,24,
%U A105022 32,16,32,32,32,1,2,4,8,4,8,12,16,4,8,12,24,12,24,24,32,4,8,12,24,12,24
%N A105022 Entries in the n-th row of Pascal's triangle that have the 2's bit set in their binary expansion.
%C A105022 0's only occur for n=0 and 1, 1's only occur for n=2^k, 2's only occur for n=1+2^k, etc. - Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 01 2005
%t A105022 PascalTwosBitCount[m_] := Plus @@ Table[BitAnd[Binomial[m, k], 2]/ 2, {k, 0, m}]
%Y A105022 Cf. A104176.
%Y A105022 Adjacent sequences: A105019 A105020 A105021 this_sequence A105023 A105024 A105025
%Y A105022 Sequence in context: A100461 A124904 A129144 this_sequence A105970 A133950 A094265
%K A105022 easy,nonn
%O A105022 0,4
%A A105022 Jacob A Siehler (siehlerj(AT)DubyaEllYou.edu), Apr 03 2005
%E A105022 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 01 2005
%I A105970
%S A105970 1,2,1,2,4,4,3,4,5,6,8,6,5,10,7,8,14,8,9,12,11,12,12,12,14,18,13,12,22,
%T A105970 16,15,24,12,18,24,18,19,20,24,20,28,22,16,28,23,24,32,20,25,38
%N A105970 Number of ordered integral solutions of Descartes' equation 2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2 with a+b+c+d=4n-2.
%C A105970 This count includes only primitive quadruples, i.e. with gcd=1. There may also be nonprimitive quads, e.g. with n=5, 4n-2=18, we have 4 primitive quads (-1,2,6,11),(-2,3,7,10),(1,1,4,12),(-3,5,8,8) and also the nonprimitive (0,3,3,12) and (3,6,6,9). - Colin Mallows (colinm(AT)avaya.com), May 11 2005
%D A105970 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks and C. Yan, Apollonian Circle Packings: Number Theory. J. Number Theory, 100 (year?), 1-45. arXiv.math.NT0009113,2003
%D A105970 J. C. Lagarias, C. L. Mallows and A. R. Wilks, Beyond the Descartes Circle Theorem, Amer. Math. Monthly, 109 (2002), 338-361.
%e A105970 a(5)=4 because we have the quadruples (1,1,4,12),(-1,2,6,11),(-2,3,7,10),(3,5,8,8).
%Y A105970 Cf. A045864.
%Y A105970 Adjacent sequences: A105967 A105968 A105969 this_sequence A105971 A105972 A105973
%Y A105970 Sequence in context: A124904 A129144 A105022 this_sequence A133950 A094265 A108802
%K A105970 nonn
%O A105970 1,2
%A A105970 Colin Mallows (colinm(AT)avaya.com), Apr 28 2005
%E A105970 More terms from Colin Mallows (colinm(AT)avaya.com), May 11 2005
%I A133950
%S A133950 1,2,1,2,4,4,4,5,6,4,5,5,4,8,10,6,6,6,6,8,8,4,8,12,6,8,11,6,8,8,8,14,8,
%T A133950 8,14,9,4,8,16,8,8,8,6,16,12,4,12,17,9,12,13,6,8,16,18,18,8,4,11,11,4,
%U A133950 12,28,20,16,8,6,13,16,8,14,14,4,12,19,14,16,8,12,31,10,4,11,22,8,8,18
%N A133950 a(n) = the number of "isolated divisors" of n(n+1)/2. A positive divisor k of n is isolated if neither k-1 nor k+1 divides n.
%H A133950 Ray Chandler, Table of n, a(n) for n=1..10000
%F A133950 a(n) = A063440(n) - A133949(n) = A132881(A000217(n)).
%e A133950 a(8)=5 because 36 (=8*9/2) has 5 isolated divisors: 6,9,12,18,36.
%p A133950 with(numtheory): b:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1,div)=false and member(div[i]+1,div)=false then ISO:= `union`(ISO,{div[i]}) end if end do end proc: seq(nops(b((1/2)*j*(j+1))),j=1..80); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 15 2007
%Y A133950 Cf. A133948, A133949, A063440.
%Y A133950 Adjacent sequences: A133947 A133948 A133949 this_sequence A133951 A133952 A133953
%Y A133950 Sequence in context: A129144 A105022 A105970 this_sequence A094265 A108802 A023673
%K A133950 nonn
%O A133950 1,2
%A A133950 Leroy Quet (qq-quet(AT)mindspring.com), Sep 30 2007
%E A133950 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 15 2007
%E A133950 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jun 23 2008
%I A094265
%S A094265 0,0,1,1,2,1,2,4,4,4,8,5,9,8,9,6,9,12,11,10,10,16,11,16,12,20,10,18,26,
%T A094265 20,25,26,23,24,26,27,31,28,25,28,26,30,25,36,25,31,37,36,25,35,36,37,
%U A094265 49,36,48,48,49,48,44,45,41,48,49,40,56,45,49,64,49,45,53,60,64,64,65
%N A094265 Largest number in n-th row of triangle A094263.
%Y A094265 Cf. A094263.
%Y A094265 Adjacent sequences: A094262 A094263 A094264 this_sequence A094266 A094267 A094268
%Y A094265 Sequence in context: A105022 A105970 A133950 this_sequence A108802 A023673 A132965
%K A094265 nonn,easy,less
%O A094265 1,5
%A A094265 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
%E A094265 Edited and extended by David Wasserman (dwasserm(AT)earthlink.net), Jan 24 2007
%I A108802
%S A108802 1,2,1,2,4,4,5,6,5,4,5,4,5,6,5,4,6,10,10,10,10,10,6,10,11,8,7,8,7,10,12,
%T A108802 6,9,10,9,10,17,14,14,14,15,16,16,14,16,14,13,14,15,20,16,16,20,22,18,24,
%U A108802 24,22,17,18,18,24,16,14,19,20,21,24,26,20,21,20,29,28,27,22,29,28,21,20
%N A108802 Self-convolution of A040051 (parity of partition function).
%Y A108802 Cf. A000041, A040051, A108803.
%Y A108802 Adjacent sequences: A108799 A108800 A108801 this_sequence A108803 A108804 A108805
%Y A108802 Sequence in context: A105970 A133950 A094265 this_sequence A023673 A132965 A022597
%K A108802 nonn
%O A108802 1,2
%A A108802 njas, Jul 09 2005
%I A023673
%S A023673 0,0,1,2,1,2,4,4,5,6,5,6,9,8,9,10,11,12,14,14,13,14,17,16,17,18,19,
%T A023673 20,23,22,23,24,25,26,27,26,27,30,30,30,31,32,33,36,35,36,37,38,39,
%U A023673 40,41,42,43,46,45,46,45,46,49,48,49,50,51,52,55,54,55,56,58,58,59
%N A023673 Convolution of A014306 with itself.
%Y A023673 Adjacent sequences: A023670 A023671 A023672 this_sequence A023674 A023675 A023676
%Y A023673 Sequence in context: A133950 A094265 A108802 this_sequence A132965 A022597 A073252
%K A023673 nonn
%O A023673 1,4
%A A023673 Clark Kimberling (ck6(AT)evansville.edu)
%I A132965
%S A132965 1,2,1,2,4,4,5,6,8,10,12,14,17,22,24,30,36,40,48,56,65,76,88,100,116,
%T A132965 134,152,174,200,226,257,292,328,372,420,472,532,598,668,750,840,936,
%U A132965 1045,1166,1296,1442,1604,1776,1972,2186,2416,2672,2952,3256,3592,3960
%N A132965 Expansion of f(-q^8) * chi(q)^2 in powers of q where f(), chi() are Ramanujan theta functions.
%F A132965 Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^8) / (eta(q)^2 * eta(q^4)^2) in powers of q.
%F A132965 Euler transform of period 8 sequence [ 2, -2, 2, 0, 2, -2, 2, -1, ...].
%F A132965 G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A132966.
%F A132965 G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 + x^(4*k)) / (1 + x^(2*k)).
%e A132965 q + 2*q^5 + q^9 + 2*q^13 + 4*q^17 + 4*q^21 + 5*q^25 + 6*q^29 + 8*q^33 + ...
%o A132965 (PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A) / eta(x + A)^2 / eta(x^4 + A)^2, n))}
%Y A132965 Adjacent sequences: A132962 A132963 A132964 this_sequence A132966 A132967 A132968
%Y A132965 Sequence in context: A094265 A108802 A023673 this_sequence A022597 A073252 A134005
%K A132965 nonn
%O A132965 0,2
%A A132965 Michael Somos, Aug 23 2007
%I A022597
%S A022597 1,2,1,2,4,4,5,6,9,12,13,16,21,26,29,36,46,54,62,74,90,106,122,142,171,
%T A022597 200,227,264,311,358,408,470,545,626,709,810,933,1062,1198,1362,1555,
%U A022597 1760,1980,2238,2536,2858,3205,3602,4063,4560,5092,5704,6400,7150,7966
%V A022597 1,-2,1,-2,4,-4,5,-6,9,-12,13,-16,21,-26,29,-36,46,-54,62,-74,90,-106,122,-142,171,
%W A022597 -200,227,-264,311,-358,408,-470,545,-626,709,-810,933,-1062,1198,-1362,1555,-1760,
%X A022597 1980,-2238,2536,-2858,3205,-3602,4063,-4560,5092,-5704,6400,-7150,7966
%N A022597 Expansion of Product (1+q^m)^(-2); m=1..inf.
%C A022597 McKay-Thompson series of class 24J for the Monster group.
%D A022597 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
%D A022597 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A022597 T. D. Noe, Table of n, a(n) for n=0..1000
%H A022597 D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
%F A022597 Expansion of q^(1/12)(eta(q)/eta(q^2))^2 in powers of q.
%F A022597 Euler transform of period 2 sequence [ -2, 0, ...]. - Michael Somos Sep 10 2005
%F A022597 Expansion of chi(-q)^2 in powers of q where chi() is a Ramanujan theta function.
%F A022597 G.f. is a period 1 Fourier series which satisfies f(-1/(288 t)) = 2 / f(t) where q = exp(2 pi i t).
%e A022597 T24J = 1/q - 2*q^11 + q^23 - 2*q^35 + 4*q^47 - 4*q^59 + 5*q^71 - 6*q^83 + ...
%o A022597 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)/eta(x^2+A))^2, n))} /* Michael Somos Sep 10 2005 */
%Y A022597 a(n)=(-1)^n*A073252(n).
%Y A022597 Convolution square of A081362. Convolution inverse of A022567.
%Y A022597 Adjacent sequences: A022594 A022595 A022596 this_sequence A022598 A022599 A022600
%Y A022597 Sequence in context: A108802 A023673 A132965 this_sequence A073252 A134005 A132320
%K A022597 sign,nice,easy
%O A022597 0,2
%A A022597 njas
%I A073252
%S A073252 1,2,1,2,4,4,5,6,9,12,13,16,21,26,29,36,46,54,62,74,90,106,122,142,171,
%T A073252 200,227,264,311,358,408,470,545,626,709,810,933,1062,1198,1362,1555,
%U A073252 1760,1980,2238,2536,2858,3205,3602,4063,4560,5092,5704,6400,7150,7966
%N A073252 Expansion of Product (1+q^(2*m-1))^2, m=1..infinity.
%C A073252 Combinatorial interpretation of sequence: [ X1,X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n)=#pairs with sum of entries = n.
%C A073252 McKay-Thompson series of class 48g for the Monster group.
%D A073252 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
%D A073252 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A073252 D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
%F A073252 G.f.: 1/(Prod_{k>0} 1+(-x)^k)^2 = (Prod_{k>0} 1+x^(2k-1))^2.
%F A073252 Expansion of q^(1/12)(eta(q^2)^2/(eta(q)eta(q^4)))^2 in powers of q.
%F A073252 Euler transform of period 4 sequence [2, -2, 2, 0, ...].
%F A073252 Expansion of chi(q)^2 in powers of q where chi() is a Ramanujan theta function.
%e A073252 a(4)=4:[ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ]
%e A073252 T48g = 1/q + 2q^11 + q^23 + 2q^35 + 4q^47 + 4q^59 + 5q^71 + 6q^83 +...
%e A073252 T48g = 1/q + 2*q^11 + q^23 + 2*q^35 + 4*q^47 + 4*q^59 + 5*q^71 + 6*q^83 +...
%o A073252 (PARI) a(n)=if(n<0,0,polcoeff(prod(i=1,(1+n)\2,1+x^(2*i-1),1+x*O(x^n))^2,n))
%o A073252 (PARI) a(n)=if(n<0,0,polcoeff(1/prod(i=1,n,1+(-x)^i,1+x*O(x^n))^2,n))
%o A073252 (PARI) {a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff( (eta(x^2+A)^2/ eta(x+A)/ eta(x^4+A))^2,n))}
%Y A073252 A022597(n)=(-1)^n*a(n).
%Y A073252 Adjacent sequences: A073249 A073250 A073251 this_sequence A073253 A073254 A073255
%Y A073252 Sequence in context: A023673 A132965 A022597 this_sequence A134005 A132320 A076369
%K A073252 nonn,easy
%O A073252 0,2
%A A073252 Michael Somos, Jul 22, 2002
%E A073252 Comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).
%I A134005
%S A134005 1,2,1,2,4,4,5,6,9,12,13,16,21,26,29,36,46,54,62,76,94,108,126,150,179,
%T A134005 210,239,282,335,384,440,512,597,684,781,902,1041,1186,1347,1544,1768,
%U A134005 2006,2268,2584,2941,3318,3742,4236,4792,5392,6053,6820,7681,8604,9632
%V A134005 1,-2,1,-2,4,-4,5,-6,9,-12,13,-16,21,-26,29,-36,46,-54,62,-76,94,-108,126,-150,179,
%W A134005 -210,239,-282,335,-384,440,-512,597,-684,781,-902,1041,-1186,1347,-1544,1768,-2006,
%X A134005 2268,-2584,2941,-3318,3742,-4236,4792,-5392,6053,-6820,7681,-8604,9632
%N A134005 Expansion of (chi(-q) * chi(-q^19))^2 in powers of q where chi() is a Ramanujan theta function.
%F A134005 Expansion of q^(5/3) * ( eta(q) * eta(q^19) / (eta(q^2) * eta(q^38)))^2 in powers of q.
%F A134005 Euler transform of period 38 sequence [ -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, ...].
%F A134005 G.f. is a period 1 Fourier series which satisfies f(-1 / (342 t)) = 4 / f(t) where q = exp(2 pi i t).
%F A134005 G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(19*k)))^-2.
%e A134005 q^-5 - 2*q^-2 + q - 2*q^4 + 4*q^7 - 4*q^10 + 5*q^13 - 6*q^16 + 9*q^19 - ...
%o A134005 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^19 + A) / eta(x^2 + A) / eta(x^38 + A) )^2, n))}
%Y A134005 A112199(n) = - A134004(n-2) - a(2*n+1) / 2. Convolution inverse of A134004.
%Y A134005 Adjacent sequences: A134002 A134003 A134004 this_sequence A134006 A134007 A134008
%Y A134005 Sequence in context: A132965 A022597 A073252 this_sequence A132320 A076369 A072727
%K A134005 sign
%O A134005 0,2
%A A134005 Michael Somos, Oct 01 2007
%I A132320
%S A132320 1,2,1,2,4,4,5,6,9,12,13,18,25,28,33,44,54,64,74,92,114,132,155,186,224,
%T A132320 260,303,360,424,488,565,662,770,888,1018,1180,1366,1560,1780,2048,2345,
%U A132320 2668,3034,3460,3946,4468,5052,5734,6502,7328,8255,9320,10512,11808
%V A132320 1,-2,1,-2,4,-4,5,-6,9,-12,13,-18,25,-28,33,-44,54,-64,74,-92,114,-132,155,-186,224,
%W A132320 -260,303,-360,424,-488,565,-662,770,-888,1018,-1180,1366,-1560,1780,-2048,2345,-2668,
%X A132320 3034,-3460,3946,-4468,5052,-5734,6502,-7328,8255,-9320,10512,-11808
%N A132320 Expansion of q^-1 * (chi(-q) * chi(-q^11))^2 in powers of q where chi() is a Ramanujan theta function.
%F A132320 Expansion of (eta(q) * eta(q^11)/( eta(q^2) * eta(q^22)))^2 in powers of q.
%F A132320 Euler transform of period 22 sequence [ -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, ...].
%F A132320 G.f. is a Fourier series which satisfies f(-1 / (22 t)) = 4 / f(t) where q = exp(2 pi i t).
%F A132320 G.f.: x^-1 * (Product_{k>0} (1+x^k) * (1+x^(11*k)))^-2.
%F A132320 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 4 * u + 4 * u * v.
%e A132320 q^-1 - 2 + q - 2*q^2 + 4*q^3 - 4*q^4 + 5*q^5 - 6*q^6 + 9*q^7 - ...
%o A132320 (PARI) {a(n) = local(A); if(n<-1, 0, n++; A = x*O(x^n); polcoeff( (eta(x+A) * eta(x^11+A) / eta(x^2+A) / eta(x^22+A))^2, n))}
%Y A132320 A058568(n) = a(n) unless n = 0.
%Y A132320 Adjacent sequences: A132317 A132318 A132319 this_sequence A132321 A132322 A132323
%Y A132320 Sequence in context: A022597 A073252 A134005 this_sequence A076369 A072727 A057061
%K A132320 sign
%O A132320 -1,2
%A A132320 Michael Somos, Aug 18 2007
%I A076369
%S A076369 2,1,2,4,4,7,6,8,9,11,10,12,12,15,16,16,16,18,18,20,22,23,22,24,25,27,
%T A076369 27,28,28,29,30,32,34,35,36,36,36,39,40,40,40,41,42,44,45,47,46,48,49,
%U A076369 50,52,52,52,54,56,56,58,59,58,60,60,63,63,64,66,65,66,68,70,69,70,72
%N A076369 n + mu(n), where mu is the Moebius-function (A008683).
%C A076369 a(n) = n iff n is not square-free:
%C A076369 a(A013929(k))=A013929(k) and a(A005117(k))<>A005117(k).
%F A076369 a(n)=n+MoebiusMu[n]=A000027[n]+A008683[n].
%Y A076369 Cf. A000027, A008683, A063015, A076840.
%Y A076369 Adjacent sequences: A076366 A076367 A076368 this_sequence A076370 A076371 A076372
%Y A076369 Sequence in context: A073252 A134005 A132320 this_sequence A072727 A057061 A034804
%K A076369 nonn
%O A076369 1,1
%A A076369 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com) and Labos E. (labos(AT)ana.sote.hu), Oct 14 2002
%I A072727
%S A072727 0,1,1,2,1,2,4,5,1,2,4,5,6,9,9,12,1,2,4,5,6,9,9,12,8,13,17,22,13,20,22,
%T A072727 29,1,2,4,5,6,9,9,12,8,13,17,22,13,20,22,29,10,17,25,32,25,38,40,53,17,
%U A072727 28,38,49,32,49,53,70
%N A072727 Denominator of the rationals >= 1 whose continued fractions consist of only even terms, in ascending order by the sum of the continued fraction terms, and descending by lowest order continued fraction terms to highest.
%F A072727 a(n) = A072726(m) where m = n - floor(log_2(n)). a(0) = 0, a(2^k) = 1, a(2^k + 1) = 2, a(2^k - 1) = the k-th Pell number.
%Y A072727 Cf. A072726, A071585, A071766, A072728, A072729, A000129.
%Y A072727 Adjacent sequences: A072724 A072725 A072726 this_sequence A072728 A072729 A072730
%Y A072727 Sequence in context: A134005 A132320 A076369 this_sequence A057061 A034804 A050042
%K A072727 easy,frac,nice,nonn
%O A072727 0,4
%A A072727 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2002
%I A057061
%S A057061 2,1,2,4,5,3,5,3,6,8,6,9,5,3,9,3,8,6,12,8,6,13,9,3,9,5,3,14,12,8,10,6,
%T A057061 17,15,5,3,15,9,5,18,12,10,20,18,14,12,21,9,5,3,21,
%U A057061 15,13,3,20,14,8,6,24,20,18,8,19,15,13,9,21,15,5,3,26,20,12,6
%N A057061 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; each k is an R(i(k),j(k)) and A057058(n)=j(n-th prime)).
%C A057061 R(i,j)=C(i+j-1,2)+i; cf. A000027, A057060.
%Y A057061 Adjacent sequences: A057058 A057059 A057060 this_sequence A057062 A057063 A057064
%Y A057061 Sequence in context: A132320 A076369 A072727 this_sequence A034804 A050042 A138256
%K A057061 nonn
%O A057061 1,1
%A A057061 Clark Kimberling (ck6(AT)evansville.edu), Jul 30 2000
%I A034804
%S A034804 0,1,0,2,1,2,4,5,6,14,17,20,48,57,68,162,193,230,548,653,778,1854,2209,
%T A034804 2632,6272,7473,8904,21218,25281,30122,71780,85525,101902,242830,
%U A034804 289329,344732,821488,978793,1166220,2779074,3311233,3945294,9401540
%N A034804 Consider the sequence of 4-tuples {0,a,b,c} (c>=a+b; a,b,c>0)) which have the smallest integer 'c' required to reach {k,k,k,k} in n steps under map {r,s,t,u}->{|r-s|,|s-t|,|t-u|,|u-r|}. This sequence gives the third term 'b' of these quadruples.
%F A034804 a(n)= 2*Trib(2*q) if r=0; Trib(2*q-1)+Trib(2*q+1) if r=1; Trib(2*q)+Trib(2*q+1) if r=2 where q=[(n-1)/3], r=n-1 (mod 3) and Trib denotes the tribonacci sequence (A000073) with Trib(-1)=1. G.f.: (-x^7+2*x^6-2*x^5+2*x^4-2*x^3-x)/(x^9+x^6+3*x^3-1). Recurrence: a(n)=3*a(n-3)+a(n-6)+a(n-9), n >= 10.
%e A034804 a(10)=14 because {0, 5, 14, 31}->{5, 9, 17, 31}->{4, 8, 14, 26}->{4, 6, 12, 22}->{2, 6, 10, 18}->{4, 4, 8, 16}->{0, 4, 8, 12}->{4, 4, 4, 12}->{0, 0, 8, 8}->{0, 8, 0, 8}->{8, 8, 8, 8} ('b'=14 in the first 4-tuple and there is no quadruple with a+b<=c<=31 and 10 steps).
%Y A034804 A034803, A045794 (or A065678) give the terms 'a' and 'c' respectively.
%Y A034804 Adjacent sequences: A034801 A034802 A034803 this_sequence A034805 A034806 A034807
%Y A034804 Sequence in context: A076369 A072727 A057061 this_sequence A050042 A138256 A139145
%K A034804 nonn
%O A034804 1,4
%A A034804 njas
%E A034804 Better description, more terms, formula, etc. from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 24 2001
%I A050042
%S A050042 1,2,1,2,4,5,7,8,10,11,13,14,16,20,25,32,40,41,43,44,46,50,55,62,70,80,
%T A050042 91,104,118,134,154,179,211,212,214,215,217,221,226,233,241,251,262,
%U A050042 275,289,305,325,350,382
%N A050042 a(n)=a(n-1)+a(m), where m=n-1-2^p, and 2^p= 4.
%Y A050042 Adjacent sequences: A050039 A050040 A050041 this_sequence A050043 A050044 A050045
%Y A050042 Sequence in context: A072727 A057061 A034804 this_sequence A138256 A139145 A058553
%K A050042 nonn
%O A050042 1,2
%A A050042 Clark Kimberling (ck6(AT)evansville.edu)
%I A138256
%S A138256 1,2,1,2,4,5,13,5,3,4,6,5,7,13,13,5,16,10,15,4,13,11,15,5,22,7,9,13,30,
%T A138256 13,33,17,11,16,13,10,19,37,7,13,8,13,55,11,14,67,35,5,43,22,82,7,108,
%U A138256 92,20,13,37,30,120,13,11,33,26,17,14,11,23,16,24,13,72,10,13,19,22,37
%N A138256 Smallest positive integer m such that n divides [3^m/m] (=A092763(m)).
%C A138256 This sequence is well-defined.
%H A138256 Romanian Master in Mathematics Contest, Bucharest, 2007, Problem 3
%Y A138256 Cf. A092763, A138255, A138257, A138258, A138259, A138260, A138261, A138262, A138263.
%Y A138256 Adjacent sequences: A138253 A138254 A138255 this_sequence A138257 A138258 A138259
%Y A138256 Sequence in context: A057061 A034804 A050042 this_sequence A139145 A058553 A112179
%K A138256 nonn
%O A138256 1,2
%A A138256 Max Alekseyev (maxal(AT)cs.ucsd.edu), Mar 09 2008.
%I A139145
%S A139145 1,1,2,1,2,4,6,1,2,4,6,16,20,36,42,1,2,4,6,16,20,36,42,256,272,400,420,
%T A139145 1296,1332,1764,1806,1,2,4,6,16,20,36,42,256,272,400,420,1296,1332,1764,
%U A139145 1806,65536,65792,73984,74256,160000,160400,176400,176820,1679616
%N A139145 a(1) = 1, a(2*n) = a(n)^2, a(2*n+1) = a(n)*(a(n)+1).
%C A139145 a(2^n - 1) = A007018(n-1).
%Y A139145 Adjacent sequences: A139142 A139143 A139144 this_sequence A139146 A139147 A139148
%Y A139145 Sequence in context: A034804 A050042 A138256 this_sequence A058553 A112179 A038067
%K A139145 nonn
%O A139145 1,3
%A A139145 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 10 2008
%I A058553
%S A058553 1,2,1,2,4,6,9,8,13,20,22,28,34
%V A058553 1,-2,1,-2,4,-6,9,-8,13,-20,22,-28,34
%N A058553 McKay-Thompson series of class 20D for Monster.
%D A058553 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058553 T20D = 1/q - 2*q + q^3 - 2*q^5 + 4*q^7 - 6*q^9 + 9*q^11 - 8*q^13 + 13*q^15 - ...
%Y A058553 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058553 Adjacent sequences: A058550 A058551 A058552 this_sequence A058554 A058555 A058556
%Y A058553 Sequence in context: A050042 A138256 A139145 this_sequence A112179 A038067 A136102
%K A058553 sign
%O A058553 -1,2
%A A058553 njas, Nov 27, 2000
%I A112179
%S A112179 1,2,1,2,4,6,9,8,13,20,22,28,34,46,57,68,87,104,127,152,187,232,267,318,
%T A112179 388,462,545,632,753,896,1043,1216,1416,1664,1928,2228,2597,2996,3454,
%U A112179 3976,4585,5286,6031,6900,7918,9060,10325,11720,13372,15228
%N A112179 McKay-Thompson series of class 40B for the Monster group.
%D A112179 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A112179 T40B = 1/q +2*q +q^3 +2*q^5 +4*q^7 +6*q^9 +9*q^11 +8*q^13 +...
%Y A112179 Adjacent sequences: A112176 A112177 A112178 this_sequence A112180 A112181 A112182
%Y A112179 Sequence in context: A138256 A139145 A058553 this_sequence A038067 A136102 A139248
%K A112179 nonn
%O A112179 0,2
%A A112179 Michael Somos, Aug 28 2005
%I A038067
%S A038067 2,1,2,4,6,9,18,34,56,99,186,344,630,1161,2182,4114,7710,
%T A038067 14532,27594,52476,99858,190557,364722,699214,1342176,2580795,
%U A038067 4971008,9587556,18512790,35790267,69273666,134219794,260300986
%V A038067 2,-1,2,-4,6,-9,18,-34,56,-99,186,-344,630,-1161,2182,-4114,7710,
%W A038067 -14532,27594,-52476,99858,-190557,364722,-699214,1342176,-2580795,
%X A038067 4971008,-9587556,18512790,-35790267,69273666,-134219794,260300986
%N A038067 Prod{k=1..inf}((1+x^k)^a(k)) = 1+2x.
%H A038067 N. J. A. Sloane, Weigh transform
%F A038067 Dirichlet convolution of A038063 with characteristic function of powers of 2.
%F A038067 a(n) = 1/n*(-(-2)^n + Sum_{dDeterminacion geometrica de los numeros primos y perfectos".
%e A139248 Triangle begins:
%e A139248 1
%e A139248 1, 2
%e A139248 1, 2, 4, 8
%e A139248 1, 2, 4, 8, 16, 32
%e A139248 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
%Y A139248 Cf. A000043, A000668, A018254, A018487, A019279, A061652, A090748, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655, A139246.
%Y A139248 Adjacent sequences: A139245 A139246 A139247 this_sequence A139249 A139250 A139251
%Y A139248 Sequence in context: A112179 A038067 A136102 this_sequence A138380 A016013 A049903
%K A139248 nonn,tabf
%O A139248 1,3
%A A139248 Omar E. Pol (info(AT)polprimos.com), Apr 22 2008
%I A138380
%S A138380 1,2,1,2,4,8,4,8,16,32,16,32,64,128,64,128,256,512,256,512,1024,2048,
%T A138380 1024,2048,4096,8192,4096,8192,16384,32768
%V A138380 1,2,-1,-2,-4,-8,4,8,16,32,-16,-32,-64,-128,64,128,256,512,-256,-512,-1024,-2048,1024,
%W A138380 2048,4096,8192,-4096,-8192,-16384,-32768
%N A138380 First differences of A138377.
%Y A138380 Adjacent sequences: A138377 A138378 A138379 this_sequence A138381 A138382 A138383
%Y A138380 Sequence in context: A038067 A136102 A139248 this_sequence A016013 A049903 A024739
%K A138380 sign
%O A138380 0,2
%A A138380 Paul Curtz (bpcrtz(AT)free.fr), May 08 2008
%I A016013
%S A016013 2,1,2,4,8,6,32,64,12,256,512,24,2048,36,30,16384,32768,96,72,
%T A016013 262144,192,1,2
%N A016013 Number of ways to write 1/n as a sum of exactly Q unit fractions.
%Y A016013 Adjacent sequences: A016010 A016011 A016012 this_sequence A016014 A016015 A016016
%Y A016013 Sequence in context: A136102 A139248 A138380 this_sequence A049903 A024739 A024959
%K A016013 nonn
%O A016013 1,1
%A A016013 Robert G. Wilson v (rgwv(AT)rgwv.com)
%I A049903
%S A049903 1,2,1,2,4,8,16,26,34,92,184,362,706,1346,2422,3860,5206,14270,28540,
%T A049903 57074,114130,228194,456118,911252,1819990,3629570,7216336,14261552,
%U A049903 27840046,52961774,95291566,151882910,204844684,561572276
%N A049903 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2n-2-2^(p+1) and 2^pComplexity of finite sequences of zeros and ones and geometry of finite spaces of function, Funct. Anal. Other. Math vol 1 iss 1 (2006) pp 1-15.
%o A135547 (C++) #include #include #include using namespace std ; int inComp( const vector< set > & comps,const int n) { for(int i=0; i < comps.size() ;i++) if ( comps[i].find(n) != comps[i].end() ) return i; return -1 ; } int firstd(const unsigned i, const int len,const unsigned allbu1, const unsigned hibit) { unsigned d= i ^ (i>>1) ; if ( (i&1) != (i & hibit) >> (len-1) ) d |= hibit ; else d &= allbu1 ; return d ; } int main(int argc, char*argv[]) { for(int n=1;; n++) { vector< set > comps ; unsigned allbu1 = 0 ; for(int i=0 ; i < n-1 ; i++) allbu1 |= (1 << i) ; const unsigned hibit = 1 <<(n-1) ; for(int i=0; i < 1< trac ; for(int ider=i;;) { int c ; if ( ( c=inComp(comps,ider) ) != -1) { comps[c].insert(ider) ; for(set::const_iterator j=trac.begin() ; j != trac.end() ; j++) comps[c].insert(*j) ; break ; } else if ( trac.find(ider) != trac.end() ) { comps.push_back(trac) ; break ; } else trac.insert(ider) ; ider= firstd(ider,n,allbu1,hibit) ; } } cout << n << " " << comps.size() <1. a(1)=2.
%o A063894 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=O(x); while(m<=n,m*=2; A=1-sqrt(1-4*x+subst(A,x,x^2))); polcoeff(A,n))
%Y A063894 Cf. A063895.
%Y A063894 Adjacent sequences: A063891 A063892 A063893 this_sequence A063895 A063896 A063897
%Y A063894 Sequence in context: A024959 A029728 A135547 this_sequence A024500 A000087 A095067
%K A063894 easy,nonn,nice
%O A063894 1,1
%A A063894 Claude Lenormand (claude.lenormand(AT)free.fr), Aug 29 2001
%I A024500
%S A024500 2,1,2,4,10,25,70,201,600,1847,5830,18778,61542,204676,689411,2347970,
%T A024500 8074762,28009676,97909318,344616322,1220539397,4347311908,15564141262,
%U A024500 55985423060,202256970300,733607297260,2670698800614,9755982909133
%N A024500 a(n) = [ C(2n,n)/n^2 ].
%Y A024500 Adjacent sequences: A024497 A024498 A024499 this_sequence A024501 A024502 A024503
%Y A024500 Sequence in context: A029728 A135547 A063894 this_sequence A000087 A095067 A032259
%K A024500 nonn
%O A024500 1,1
%A A024500 Clark Kimberling (ck6(AT)evansville.edu)
%I A000087 M1240 N0474
%S A000087 2,1,2,4,10,37,138,628,2972,14903,76994,409594,2222628,12281570,
%T A000087 68864086,391120036,2246122574,13025721601,76194378042,449155863868,
%U A000087 2666126033850,15925105028685,95664343622234,577651490729530
%N A000087 Number of rooted planar maps.
%C A000087 The number of unrooted non-separable n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005
%D A000087 W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
%D A000087 V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%H A000087 T. D. Noe, Table of n, a(n) for n=1..200
%H A000087 V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.
%F A000087 a(n)=(1/3n)[(n+2)binomial(3n, n)/((3n-2)(3n-1)) + Sum_{0C-program for computing the initial terms of this sequence
%H A095067 Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]
%Y A095067 a(n) = A095060(n)-A095062(n).
%Y A095067 Adjacent sequences: A095064 A095065 A095066 this_sequence A095068 A095069 A095070
%Y A095067 Sequence in context: A063894 A024500 A000087 this_sequence A032259 A109851 A011404
%K A095067 nonn
%O A095067 1,4
%A A095067 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004
%I A032259
%S A032259 2,1,2,4,14,50,194,750,2934,11557,46010,184784,748678,3056308,
%T A032259 12562908,51952261,216011544,902512979,3787287378,15955748048,
%U A032259 67462625834,286172617162,1217561851258,5194521586232
%N A032259 Number of series-reduced dyslexic planted compound windmills with n leaves of 2 colors with no symmetries.
%H A032259 C. G. Bower, Transforms (2)
%H A032259 Index entries for sequences related to mobiles
%F A032259 Doubles (index 2+) under "DHK" (necklace, identity, unlabeled) transform.
%Y A032259 Adjacent sequences: A032256 A032257 A032258 this_sequence A032260 A032261 A032262
%Y A032259 Sequence in context: A024500 A000087 A095067 this_sequence A109851 A011404 A002211
%K A032259 nonn
%O A032259 1,1
%A A032259 Christian G. Bower (bowerc(AT)usa.net)
%I A109851
%S A109851 1,2,1,2,4,14,74,494,3854,34094,336494,3662894,43579694,562498094,
%T A109851 7827355694,116800219694,1860366043694,31500985051694,565032127195694,
%U A109851 10702123827931694,213443957842651694,4471022472151771694,98137749786952411694,2252472478027367131694
%N A109851 a(1) = 1, a(2) = 2; for n > 2, sum of absolute differences of all combinations of pairs of previous terms.
%C A109851 The final digit approaches a limit. That is, after the first few terms, all the terms end in 70194710743368411694, and as more terms go by, even more digits remain constant. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 04 2006
%F A109851 a(n+1) = a(n) + sum [ absolute{a(n) - a(k)}, k = 1 to n].
%e A109851 14 is the next term after 4= a(5), 14 = 4 + abs(4-a(1)) + abs(4-a(2)) + abs( 4-a(3)) + abs(4-a(4)).
%Y A109851 Adjacent sequences: A109848 A109849 A109850 this_sequence A109852 A109853 A109854
%Y A109851 Sequence in context: A000087 A095067 A032259 this_sequence A011404 A002211 A132309
%K A109851 nonn
%O A109851 0,2
%A A109851 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 06 2005
%E A109851 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 04 2006
%I A011404
%S A011404 1,2,1,2,5,0,8,6,1,6,9,0,0,6,2,1,9,0,9,9,2,4,8,7,3,6,8,7,8,5,6,8,9,
%T A011404 3,1,6,4,9,3,5,8,5,8,0,9,8,9,5,3,9,1,9,3,5,6,7,7,7,1,0,3,1,3,1,7,2,
%U A011404 9,1,1,1,7,8,3,4,7,3,2,3,1,8,5,5,5,4,9,8,0,3,3,5,4,3,9,9,1,2,3,9,3
%N A011404 Decimal expansion of 15th root of 18.
%Y A011404 Adjacent sequences: A011401 A011402 A011403 this_sequence A011405 A011406 A011407
%Y A011404 Sequence in context: A095067 A032259 A109851 this_sequence A002211 A132309 A122881
%K A011404 nonn,cons
%O A011404 1,2
%A A011404 njas
%I A002211 M0118 N0047
%S A002211 2,1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,3,1,1,1,90,2,1,12,1,1,1,1,
%T A002211 5,2,6,1,6,3,1,1,2,5,2,1,2,1,1,4,1,2,2,3,2,1,1,4,1,1,2,5,2,1,1,3,29,
%U A002211 8,3,1,4,3,1,10,50,1,2,2,7,6,2,2,16,4,4,2,2,3,1,1,7,1,5,1,2,1,5,3,1
%N A002211 Continued fraction for Khintchine's constant.
%D A002211 D. Shanks, Further evaluation of Khintchine's constant, Math. Comp., 14 (1960), 370-371.
%D A002211 D. Shanks and J. W. Wrench, Jr., Khintchine's constant, Amer. Math. Monthly, 66 (1959), 276-279.
%D A002211 J. W. Wrench, Jr. and D. Shanks, Questions concerning Khintchine's constant and the efficient computation of regular continued fractions, Math. Comp., 20 (1966), 444-448.
%H A002211 T. D. Noe, Table of n, a(n) for n=1..1000
%H A002211 G. Xiao, Contfrac
%H A002211 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%H A002211 Index entries for continued fractions for constants
%H A002211 H. Havermann, Simple Continued Fraction Expansion of Khinchin's Constant
%t A002211 ContinuedFraction[ Khinchin, 100]
%Y A002211 Cf. A002210.
%Y A002211 Adjacent sequences: A002208 A002209 A002210 this_sequence A002212 A002213 A002214
%Y A002211 Sequence in context: A032259 A109851 A011404 this_sequence A132309 A122881 A135506
%K A002211 cofr,nonn,nice,easy
%O A002211 1,1
%A A002211 njas
%E A002211 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 31 2001
%I A132309
%S A132309 1,2,1,2,5,1,2,5,8,1,2,9,9,11,1,2,9,24,14,14,1,2,13,25,50,20,17,1,2,13,
%T A132309 48,55,90,27,20,1,2,17,49,133,105,147,35,23,1,2,17,80,140,308,182,224,
%U A132309 44,26,1
%N A132309 A007318^(-1) * A132307.
%C A132309 Row sums = A077552: (1, 3, 8, 16, 32, 64, 128,...).
%F A132309 Inverse binomial transform of triangle A132307
%e A132309 First few rows of the triangle are:
%e A132309 1;
%e A132309 2, 1;
%e A132309 2, 5, 1;
%e A132309 2, 5, 8, 1;
%e A132309 2, 9, 9, 11, 1;
%e A132309 2, 9, 24, 14, 14, 1;
%e A132309 2, 13, 25, 50, 20, 17, 1;
%e A132309 2, 13, 48, 55, 90, 27, 20, 1;
%e A132309 ...
%Y A132309 Cf. A132307, A077552.
%Y A132309 Adjacent sequences: A132306 A132307 A132308 this_sequence A132310 A132311 A132312
%Y A132309 Sequence in context: A109851 A011404 A002211 this_sequence A122881 A135506 A068822
%K A132309 nonn,tabl
%O A132309 0,2
%A A132309 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 18 2007
%I A122881
%S A122881 1,1,2,1,2,5,1,2,5,13,1,2,5,14,34,1,2,5,14,42,89,1,2,5,14,42,131,233,1,
%T A122881 2,5,14,42,132,417,610,1,2,5,14,42,132,429,1341,1597,1,2,5,14,42,132,
%U A122881 429,1429,4334,4181
%N A122881 Triangle read by rows: number of Catalan paths of 2n steps of all values less than or equal to m.
%C A122881 Convergents of k-th diagonals relate to (2k+3)-Polygons; e.g. right border relates to the Pentagon (N=5), next border relates to the Heptagon.(N=7). Convergents of the diagonals are 2 + 2Cos 2Pi/N, and are roots to Morgan-Voyce polynomials. k2 diagonal = A080937, number of Catalan paths of 2n steps of all values less than or equal to 5. k3 diagonal = A080938, number of Catalan paths of 2n steps of all values less than or equal to 7.
%F A122881 Begin with polygonal matrices of the form (exemplified by the Heptagonal matrix M3: [1, 1, 1; 1, 1, 0; 1, 0, 0]). Let matrix P3 = 1 / M3^2; then for n X n matrices P2, P3, P4...perform P^n * [1, 0, 0] letting this vector = k-th diagonal of the triangle.
%e A122881 For the right border, odd indexed Fibonacci numbers (1, 2, 5, 13, 34...), we begin with (M2) = [1, 1; 1, 0], then P2 = [1, -1; -1, 2] = 1/(M2)^2. Performing (P2)^n * [1,0] we extract the left vector (1, 2, 5, 13...), making it the right border of the triangle, k1 diagonal.
%e A122881 For the next diagonal going to the left, we begin with the Heptagonal matrix M3 = [1, 1, 1; 1, 1, 0; 1, 0, 0], take the inverse square (P3) and then perform the analogous operation getting 1, 2, 5, 14, 42...
%e A122881 First few rows of the triangle are:
%e A122881 1;
%e A122881 1, 2;
%e A122881 1, 2, 5;
%e A122881 1, 2, 5, 13;
%e A122881 1, 2, 5, 14, 34;
%e A122881 1, 2, 5, 14, 42, 89;
%e A122881 1, 2, 5, 14, 42, 131, 233;
%e A122881 1, 2, 5, 14, 42, 132, 417, 610;
%e A122881 ...
%Y A122881 Cf. A112880, A001519, A000108, A080937, A080938.
%Y A122881 Adjacent sequences: A122878 A122879 A122880 this_sequence A122882 A122883 A122884
%Y A122881 Sequence in context: A011404 A002211 A132309 this_sequence A135506 A068822 A090079
%K A122881 nonn,tabl
%O A122881 1,3
%A A122881 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 16 2006
%I A135506
%S A135506 2,1,2,5,1,7,1,1,5,11,1,13,1,5,1,17,1,19,1,1,11,23,1,5,13,1,1,29,1,31,1,
%T A135506 11,17,1,1,37,1,13,1,41,1,43,1,1,23,47,1,1,1,17,13,53,1,1,1,1,29,59,1,
%U A135506 61,1,1,1,13,1,67,1,23,1,71,1,73,1,1,1,1,13,79,1,1,41,83,1,1,43,29,1,89
%N A135506 a(n)=x(n+1)/x(n)-1 where x(1)=1 and x(k)=x(k-1)+lcm(x(k-1),k).
%C A135506 This sequence has fascinating properties related to primes. For instance : terms consist of 1's or primes only, if 3 never occurs, any prime p occur finitely many times...
%D A135506 Benoit Cloitre, Beyond Rowland's gcd sequence, in preparation, 2008
%o A135506 (PARI) x1=1;for(n=2,40,x2=x1+lcm(x1,n);t=x1;x1=x2;print1(x2/t-1,","))
%Y A135506 Cf. A106108.
%Y A135506 Adjacent sequences: A135503 A135504 A135505 this_sequence A135507 A135508 A135509
%Y A135506 Sequence in context: A002211 A132309 A122881 this_sequence A068822 A090079 A121487
%K A135506 nonn
%O A135506 1,1
%A A135506 Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 09 2008
%I A068822
%S A068822 1,2,1,2,5,2,1,2,1,10,1,4,1,2,5,4,1,2,1,20,1,2,1,4,25,2,1,4,1,10,1,4,1,
%T A068822 2,5,4,1,2,1,20,1,2,1,4,5,2,1,4,1,50,1,4,1,2,5,4,1,2,1,20,1,2,1,4,5,2,
%U A068822 1,4,1,10,1,4,1,2,25,4,1,2,1,20,1,2
%N A068822 a(n) = GCD(n,c(n)), where c(n) is the 10's complement of n.
%e A068822 a(45) = 5 as 10's complement of 45 is 100-45 = 55 and (45,55) = 5.
%Y A068822 Adjacent sequences: A068819 A068820 A068821 this_sequence A068823 A068824 A068825
%Y A068822 Sequence in context: A132309 A122881 A135506 this_sequence A090079 A121487 A057031
%K A068822 easy,nonn
%O A068822 1,2
%A A068822 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 08 2002
%I A090079
%S A090079 0,1,2,1,2,5,2,1,2,5,10,5,2,5,2,1,2,5,10,5,10,21,10,5,2,5,10,5,2,5,2,1,
%T A090079 2,5,10,5,10,21,10,5,10,21,42,21,10,21,10,5,2,5,10,5,10,21,10,5,2,5,10,
%U A090079 5,2,5,2,1,2,5,10,5,10,21,10,5,10,21,42,21,10,21,10,5,10,21,42,21
%N A090079 In binary expansion of n: reduce contiguous blocks of 0's to 0 and contiguous blocks of 1's to 1.
%C A090079 a(a(n))=a(n); a(n)=A090078(A090077(n))=A090077(A090078(n)).
%H A090079 Index entries for sequences related to binary expansion of n
%e A090079 100 -> '1100100' -> [11][00][1][00] -> [1][0][1][0] -> '1010' ->
%e A090079 10=a(100).
%Y A090079 Cf. A007088, A090077, A090078, A090080.
%Y A090079 Adjacent sequences: A090076 A090077 A090078 this_sequence A090080 A090081 A090082
%Y A090079 Sequence in context: A122881 A135506 A068822 this_sequence A121487 A057031 A078391
%K A090079 nonn,base
%O A090079 0,3
%A A090079 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2003
%I A121487
%S A121487 1,1,1,2,1,2,5,2,1,5,13,5,2,1,13,34,13,5,2,1,34,89,34,13,5,2,1,89,233,
%T A121487 89,34,13,5,2,1,233,610,233,89,34,13,5,2,1,610,1597,610,233,89,34,13,5,
%U A121487 2,1,1597,4181,1597,610,233,89,34,13,5,2,1,4181,10946,4181,1597,610,233
%N A121487 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having abscissa of first return equal to 2k (1<=k<=n). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
%C A121487 Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,1)=T(n,n)=fibonacci(2n-3)=A001519(n-1) for n>=2.
%D A121487 E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
%F A121487 T(n,k)=fibonacci(2n-2k-1) if k0 and k>0, A(n,k) = Sum(i=0..n-1,j=0..k-1, A[i,j]*A[n-i,k-j]). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Dec 30 2007
%e A078391 Rows start: 1; 1,1; 2,1,2; 5,2,2,5; 14,5,4,5,14; 42,14,10,10,14,42; 132,42,28,25,28,42,132; etc.
%Y A078391 Row sums are Catalan numbers A000108 (offset). Central diagonal is A001246. Cf. A067804.
%Y A078391 Adjacent sequences: A078388 A078389 A078390 this_sequence A078392 A078393 A078394
%Y A078391 Sequence in context: A090079 A121487 A057031 this_sequence A109631 A095149 A064192
%K A078391 nonn,tabl
%O A078391 0,4
%A A078391 Henry Bottomley (se16(AT)btinternet.com), Dec 24 2002
%I A109631
%S A109631 1,2,1,2,5,2,3,2,1,5,12,2,22,3,5,4,15,2,24,5,3,12,31,2,10,22,4,3,13,5,
%T A109631 38,6,12,15,5,2,26,24,22,5,18,3,14,12,5,31,53,4,16,10,15,22,47,4,12,3,
%U A109631 24,13,41,5,39,38,3,6,22,12,33,15,31,5,29,2,27,26,10,24,12,22,21,5,7,18
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