The Database of Integer Sequences, Part 6 

     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:
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(start)



%I A001826
%S A001826 1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,2,2,1,2,2,1,1,1,3,2,2,1,2,2,1,1,2,2,2,
%T A001826 2,2,1,2,2,2,2,1,1,4,1,1,1,2,3,2,2,2,2,2,1,2,2,1,2,2,1,3,1,4,2,1,2,2,2,
%U A001826 1,2,2,2,3,1,2,2,1,2,3,2,1,2,4,1,2,1,2,4,2,1,2,1,2,1,2,2,3,3,2,2,1,2,4
%N A001826 Number of divisors of n of form 4k+1.
%H A001826 Nick Hobson, <a href="http://www.research.att.com/~njas/sequences/b001826.txt">Table of n, a(n) for n = 1..10000</a>
%H A001826 Michael Gilleland, <a href="http://www.research.att.com/~njas/sequences/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%F A001826 G.f.: Sum_{n>0} x^n/(1-x^(4n)) = Sum x^(4n+1)/(1-x^(4n+1)), n=0..inf.
%F A001826 a(n) = A001227(n) - A001842(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2006
%p A001826 d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; # no. of divisors i of n with i == r mod m
%o A001826 (PARI) a(n)=if(n<1,0,sumdiv(n,d,d%4==1))
%Y A001826 Sequence in context: A049419 A046951 A050377 this_sequence A003641 A025890 A043277
%Y A001826 Adjacent sequences: A001823 A001824 A001825 this_sequence A001827 A001828 A001829
%K A001826 nonn
%O A001826 1,5
%A A001826 njas
%E A001826 Better definition from Michael Somos, Apr 26 2004

%I A003641 M0061
%S A003641 1,1,1,2,1,1,1,2,2,1,1,2,2,1,1,1,1,1,2,2,2,1,1,2,2,2,2,1,1,1,2,1,2,2,1,
%T A003641 1,1,2,2,1,4,1,2,1,2,2,1,1,4,2,1,2,1,4,2,1,2,1,1,2,2,2,2,2,1,1,2,2,2,1,
%U A003641 2,2,1,2,1,1,2,1,1,2,2,4,2,2,2,2,1,2,1,4,1,1,2,2,4,1,1,2,1,4,1,1,1,1,2
%N A003641 Number of genera of quadratic field with discriminant -4n+1.
%D A003641 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
%H A003641 <a href="http://www.research.att.com/~njas/sequences/Sindx_Qua.html#quadfield">Index entries for sequences related to quadratic fields</a>
%Y A003641 Cf. A003642.
%Y A003641 Sequence in context: A046951 A050377 A001826 this_sequence A025890 A043277 A025864
%Y A003641 Adjacent sequences: A003638 A003639 A003640 this_sequence A003642 A003643 A003644
%K A003641 nonn
%O A003641 1,4
%A A003641 njas, Mira Bernstein

%I A025890
%S A025890 1,0,0,0,0,1,0,0,1,0,1,0,1,1,0,1,1,1,1,0,2,1,1,1,2,2,1,
%T A025890 1,2,2,2,1,3,2,2,2,3,3,2,2,4,3,3,2,4,4,3,3,5,4,4,3,5,5,
%U A025890 4,4,6,5,5,4,7,6,5,5,7,7,6,5,8,7,7,6,9,8,7,7,9,9,8,7
%N A025890 Expansion of 1/((1-x^5)(1-x^8)(1-x^12)).
%Y A025890 Sequence in context: A050377 A001826 A003641 this_sequence A043277 A025864 A070242
%Y A025890 Adjacent sequences: A025887 A025888 A025889 this_sequence A025891 A025892 A025893
%K A025890 nonn
%O A025890 0,21
%A A025890 njas

%I A043277
%S A043277 1,1,1,2,1,1,1,2,2,1,1,2,3,2,1,1,2,2,1,1,1,2,1,2,2,3,3,2,2,1,
%T A043277 2,1,1,1,2,2,2,2,3,4,3,2,2,2,2,1,1,1,2,1,2,2,3,3,2,2,1,2,1,1,
%U A043277 1,2,2,1,1,2,3,2,1,1,2,2,2,2,2,2,2,3,3,4,4,3,3,2,2,2,2,2,2,2
%N A043277 Maximal run length in base 3 representation of n.
%Y A043277 Sequence in context: A001826 A003641 A025890 this_sequence A025864 A070242 A037226
%Y A043277 Adjacent sequences: A043274 A043275 A043276 this_sequence A043278 A043279 A043280
%K A043277 nonn,base
%O A043277 1,4
%A A043277 Clark Kimberling (ck6(AT)evansville.edu)

%I A025864
%S A025864 1,0,0,0,1,1,0,0,1,1,1,0,2,1,1,1,2,2,1,1,3,2,2,1,4,3,2,
%T A025864 2,4,4,3,2,5,4,4,3,6,5,4,4,7,6,5,4,8,7,6,5,9,8,7,6,10,9,
%U A025864 8,7,11,10,9,8,13,11,10,9,14,13,11,10,15,14,13,11,17,15
%N A025864 Expansion of 1/((1-x^4)(1-x^5)(1-x^12)).
%Y A025864 Sequence in context: A003641 A025890 A043277 this_sequence A070242 A037226 A089641
%Y A025864 Adjacent sequences: A025861 A025862 A025863 this_sequence A025865 A025866 A025867
%K A025864 nonn
%O A025864 0,13
%A A025864 njas

%I A070242
%S A070242 1,1,1,1,1,2,1,1,1,2,2,1,1,3,3,2,2,1,1,3,2,1,3,3,2,3,1,2,1,5,2,1,3,2,3,
%T A070242 1,1,3,2,3,3,4,1,3,1,5,3,2,1,1,5,2,2,4,5,4,2,3,3,6,1,4,2,1,3,5,1,2,4,5,
%U A070242 5,1,1,2,2,2,4,6,2,2,1,2,3,3,2,2,4,4,3,3,1,6,2,5,4,6,2,1,2,1,1,5,2,2,5
%N A070242 Card( k>0 : sigma(k)=sigma(n) ).
%o A070242 (PARI) for(n=1,150,print1(sum(i=1,10*n,if(sigma(n)-sigma(i),0,1)),","))
%Y A070242 Sequence in context: A025890 A043277 A025864 this_sequence A037226 A089641 A086995
%Y A070242 Adjacent sequences: A070239 A070240 A070241 this_sequence A070243 A070244 A070245
%K A070242 easy,nonn
%O A070242 1,6
%A A070242 Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2002

%I A037226
%S A037226 1,1,1,2,1,1,1,2,2,1,2,2,1,1,1,6,2,2,1,2,2,3,2,2,2,4,1,2,
%T A037226 2,1,1,6,4,1,2,2,8,2,2,2,1,1,8,2,8,6,6,2,2,2,1,2,4,1,3,
%U A037226 2,4,2,6,4,1,4,1,18,6,1,6,2,2,1,2,2,4,2,1,10,4,6,3,2,4
%N A037226 phi(2n+1) / multiplicative order of 2 mod 2n+1.
%C A037226 Number of primitive irreducible factors of x^(2n+1) - 1 over integers mod 2. There are no primitive irreducible factors for x^(2n)-1 because it always has the same factors as x^n-1. Considering that A000374 also counts the cycles in the map f(x) = 2x mod n, a(n) is also the number of primitive cycles of that mapping. - T. D. Noe (noe(AT)sspectra.com), Aug 01 2003
%C A037226 Equals number of irreducible factors of the cyclotomic polynomial Phi(2n+1,x) over Z/2Z. All factors have the same degree. - T. D. Noe, Mar 01 2008
%H A037226 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b037226.txt">Table of n, a(n) for n=0..10000</a>
%F A037226 a(n) = Sum{d|2n+1} mu((2n+1)/d) A000374(d), the inverse Mobius transform of A000374 - T. D. Noe (noe(AT)sspectra.com), Aug 01 2003
%Y A037226 A037225[ n ]/A002326[ n ].
%Y A037226 Cf. A000374 (number of irreducible factors of x^n - 1 over integers mod 2), A081844.
%Y A037226 Sequence in context: A043277 A025864 A070242 this_sequence A089641 A086995 A135230
%Y A037226 Adjacent sequences: A037223 A037224 A037225 this_sequence A037227 A037228 A037229
%K A037226 nonn
%O A037226 0,4
%A A037226 njas

%I A089641
%S A089641 1,2,1,1,1,2,2,1,2,2,2,1,1,2,1,2,4,2,2,3,1,4,2,5,3,3,2,2,2,1,2,2,4,1,2,
%T A089641 4,2,4,1,1,4,1,2,2,3,2,2,1,2,2,4,2,4,8,1,1,5,2,4,2,8,6,5,2,1,2,2,3,2,3,
%U A089641 4,1,3,1,2,2,7,5,2,1,2,2,2,8,2,2,4,2,1,5,2,4,4,4,2,6,2,8,2,6,6,1,2,2,2
%N A089641 Number of k, 1<=k<=n, such that the number of elements in the continued fraction for n/k is maximum.
%o A089641 (PARI) a(n)=sum(s=1,n,if(length(contfrac(n/s))-vecmax(vector(n,i,length(contfrac(n/i)))),0,1))
%Y A089641 Cf. A084242.
%Y A089641 Sequence in context: A025864 A070242 A037226 this_sequence A086995 A135230 A117957
%Y A089641 Adjacent sequences: A089638 A089639 A089640 this_sequence A089642 A089643 A089644
%K A089641 nonn
%O A089641 1,2
%A A089641 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 01 2004

%I A086995
%S A086995 2,1,1,1,2,2,1,2,2,2,2,1,2,3,2,2,1,2,1,1,2,1,2,1,2,2,1,2,1,3,2,2,0,1,1,
%T A086995 1,1,1,2,3,1,2,2,2,2,2,2,3,2,1,0,2,1,1,0,2,3,1,2,1,1,2,2,1,2,0,3,1,1,2,
%U A086995 1,0,2,1,1,2,1,0,1,2,2,1,2,1,1,0,2,1,1,1,2,2,1,1,1,1,3,0,2,3,2,1,1,1,1
%N A086995 Number of 1's in binary representation of n-th decimal digit in expansion of Pi.
%Y A086995 Cf. A000796, A086994.
%Y A086995 Sequence in context: A070242 A037226 A089641 this_sequence A135230 A117957 A139632
%Y A086995 Adjacent sequences: A086992 A086993 A086994 this_sequence A086996 A086997 A086998
%K A086995 nonn,base
%O A086995 11,1
%A A086995 Cino Hilliard (hillcino368(AT)gmail.com), Sep 23 2003
%E A086995 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 29 2003

%I A135230
%S A135230 1,1,1,2,1,1,1,2,2,1,2,2,4,3,1,1,3,6,7,4,1,2,3,9,13,11,5,1,1,4,12,22,24,
%T A135230 16,6,1,2,4,16,34,46,40,22,7,1,1,5,20,50,80,86,62,29,8,1
%N A135230 A103451 * A000012(signed) * A007318.
%C A135230 row sums = A135231
%F A135230 A103451 * A000012(signed) * A007318, where A000012(signed) = (1; -1,1; 1,-1,1;...).
%e A135230 First few rows of the triangle are:
%e A135230 1;
%e A135230 1, 1;
%e A135230 2, 1, 1;
%e A135230 1, 2, 2, 1;
%e A135230 2, 2, 4, 3, 1;
%e A135230 1, 3, 6, 7, 4, 1;
%e A135230 2, 3, 9, 13, 11, 5, 1;
%e A135230 1, 4, 12, 22, 24, 16, 6, 1;
%e A135230 2, 4, 16, 34, 46, 40, 22, 7, 1;
%e A135230 ...
%Y A135230 Cf. A103451, A135231, A007318, A000012.
%Y A135230 Sequence in context: A037226 A089641 A086995 this_sequence A117957 A139632 A029339
%Y A135230 Adjacent sequences: A135227 A135228 A135229 this_sequence A135231 A135232 A135233
%K A135230 nonn,tabl
%O A135230 1,4
%A A135230 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007

%I A117957
%S A117957 1,0,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,2,1,1,1,2,2,1,2,3,3,2,2,4,4,3,3,5,
%T A117957 6,5,4,6,8,7,6,8,10,10,9,10,13,13,12,14,17,18,16,18,22,23,22,23,28,31,
%U A117957 29,30,36,39,39,39,45,51,50,51,57,64,65,65,73,81,83,84,91,102,106,106
%N A117957 Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.
%C A117957 Also number of partitions of n such that 2k and 2k+1 occur with the same multiplicities. Example: a(26)=3 because we have [11,10,3,2], [9,8,5,4], and [7,7,6,6]. It is easy to find a bijection between these partitions and those described in the definition.
%F A117957 G.f.=1/product(1-x^(4i+1), i=1..infinity).
%e A117957 a(26)=3 because we have [21,5],[17,9], and [13,13].
%p A117957 g:=1/product(1-x^(4*i+1),i=1..50): gser:=series(g,x=0,93): seq(coeff(gser,x,n),n=0..88);
%Y A117957 Cf. A035451, A035462.
%Y A117957 Sequence in context: A089641 A086995 A135230 this_sequence A139632 A029339 A029364
%Y A117957 Adjacent sequences: A117954 A117955 A117956 this_sequence A117958 A117959 A117960
%K A117957 nonn
%O A117957 0,19
%A A117957 Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 05 2006

%I A139632
%S A139632 1,1,0,1,1,0,0,1,1,1,1,1,2,1,1,1,2,2,1,2,3,3,2,3,4,3,2,4,5,4,4,5,6,6,5,
%T A139632 6,8,7,6,8,11,10,8,11,13,11,10,13,16,15,14,17,20,18,17,20,24,23,21,25,
%U A139632 31,29,26,32,37,34,32,39,44,42,41,47,54,52,49,56,64,62,59,68,79,77,72
%N A139632 Expansion of chi(q) * chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
%F A139632 Expansion of q^(1/4) * eta(q^2)^2 * eta(q^5) / (eta(q) * eta(q^4) * eta(q^10)) in powers of q.
%F A139632 G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A139631.
%F A139632 G.f.: Product_{k>0} (1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*k))).
%e A139632 1/q + q^3 + q^11 + q^15 + q^27 + q^31 + q^35 + q^39 + q^43 + 2*q^47 + ...
%o A139632 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^5 + A) / eta(x + A) / eta(x^4 + A) / eta(x^10 + A), n))}
%Y A139632 A139631(n) = a(2*n).
%Y A139632 Sequence in context: A086995 A135230 A117957 this_sequence A029339 A029364 A122586
%Y A139632 Adjacent sequences: A139629 A139630 A139631 this_sequence A139633 A139634 A139635
%K A139632 nonn
%O A139632 0,13
%A A139632 Michael Somos, Apr 27 2008

%I A029339
%S A029339 1,0,0,0,1,1,0,0,2,1,1,1,2,2,1,2,4,2,2,3,5,4,3,4,7,5,5,
%T A029339 6,8,7,7,8,11,9,9,11,13,12,12,13,17,15,15,17,20,19,19,20,
%U A029339 25,23,23,25,29,28,28,30,35,33,33,36,41,39,39,42,48,46
%N A029339 Expansion of 1/((1-x^4)(1-x^5)(1-x^8)(1-x^11)).
%Y A029339 Sequence in context: A135230 A117957 A139632 this_sequence A029364 A122586 A079487
%Y A029339 Adjacent sequences: A029336 A029337 A029338 this_sequence A029340 A029341 A029342
%K A029339 nonn
%O A029339 0,9
%A A029339 njas

%I A029364
%S A029364 1,0,0,0,1,0,0,1,1,1,0,2,1,1,1,2,2,1,3,2,3,2,4,3,3,4,4,
%T A029364 5,4,6,5,6,6,7,7,7,9,8,9,9,11,10,11,12,13,13,13,15,15,16,
%U A029364 16,18,18,19,20,21,22,22,24,24,26,26,28,29,30,31,32,34
%N A029364 Expansion of 1/((1-x^4)(1-x^7)(1-x^9)(1-x^11)).
%Y A029364 Sequence in context: A117957 A139632 A029339 this_sequence A122586 A079487 A069010
%Y A029364 Adjacent sequences: A029361 A029362 A029363 this_sequence A029365 A029366 A029367
%K A029364 nonn
%O A029364 0,12
%A A029364 njas

%I A122586
%S A122586 1,2,1,1,1,2,2,2,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,
%T A122586 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A122586 2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A122586 Digits 1 and 2 appear alternately and each time in runs whose lengths are the powers of 3.
%F A122586 a(n)=floor(n/(3^floor(log[3](n))))
%e A122586 a(1) = 1/(3^0) = 1
%p A122586 seq( evalf(floor(n/ (3^floor(log[3](n))))), n=1..500);
%Y A122586 Sequence in context: A139632 A029339 A029364 this_sequence A079487 A069010 A087048
%Y A122586 Adjacent sequences: A122583 A122584 A122585 this_sequence A122587 A122588 A122589
%K A122586 easy,nonn
%O A122586 1,2
%A A122586 Peter C. Heinig (algorithms(AT)gmx.de), Oct 20 2006

%I A079487
%S A079487 1,1,1,1,1,1,1,2,1,1,1,2,2,2,1,1,3,3,3,2,1,1,3,4,5,4,3,1,1,4,6,7,7,
%T A079487 5,3,1,1,4,7,10,11,10,7,4,1,1,5,10,14,17,16,13,8,4,1
%N A079487 Triangle read by rows giving Whitney numbers T(n,k) of Fibonacci lattices.
%C A079487 Row sums are Fibonacii numbers A000045. - Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 07 2006
%C A079487 This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), May 07 2008
%D A079487 E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
%F A079487 Define polynomials by: if k is odd then p(k, x) = x*p(k - 1, x) + p(k - 2, x); if k is even then: p(k, x) = p(k - 1, x) + x^2*p(k - 2, x). Triangle gives array of coefficients. - Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 07 2006
%e A079487 Triangle begins:
%e A079487 {1},
%e A079487 {1, 1},
%e A079487 {1, 1, 1},
%e A079487 {1, 2, 1, 1},
%e A079487 {1, 2, 2, 2, 1},
%e A079487 {1, 3, 3, 3, 2, 1},
%e A079487 {1, 3, 4, 5, 4, 3, 1},
%e A079487 {1, 4, 6, 7, 7, 5, 3, 1},
%e A079487 {1, 4, 7, 10, 11, 10, 7, 4, 1},
%e A079487 {1, 5, 10, 14, 17, 16, 13, 8, 4, 1},
%e A079487 {1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}
%t A079487 p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = If[Mod[k, 2] == 1, x*p[k - 1, x] + p[k - 2, x], p[k - 1, x] + x^2*p[k - 2, x]]; Table[Expand[p[n, x]], {n, 0, 10}] Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[n, x], x]]}], {n, 0, 15}] w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w] - Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 07 2006
%Y A079487 Largest element in each row gives A077419.
%Y A079487 Sequence in context: A029339 A029364 A122586 this_sequence A069010 A087048 A109700
%Y A079487 Adjacent sequences: A079484 A079485 A079486 this_sequence A079488 A079489 A079490
%K A079487 nonn,tabl
%O A079487 0,8
%A A079487 njas, Jan 19 2003

%I A069010
%S A069010 0,1,1,1,1,2,1,1,1,2,2,2,1,2,1,1,1,2,2,2,2,3,2,2,1,2,2,2,1,2,1,1,1,2,2,
%T A069010 2,2,3,2,2,2,3,3,3,2,3,2,2,1,2,2,2,2,3,2,2,1,2,2,2,1,2,1,1,1,2,2,2,2,3,
%U A069010 2,2,2,3,3,3,2,3,2,2,2,3,3,3,3,4,3,3,2,3,3,3,2,3
%N A069010 Number of runs of 1's in binary representation of n.
%H A069010 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A069010 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%F A069010 a(n) =ceiling[A005811(n)/2] =A005811(n)-A033264(n). If 2^k<=n<3*2^(k-1) then a(n)=a(n-2^k)+1; if 3*2^(k-1)<=n<2^(k+1) then a(n)=a(n-2^k).
%F A069010 a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 20 2003
%F A069010 G.f.: 1/(1-x) * sum(k>=0, t/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 07 2003
%F A069010 a(n) = A000120(n) - A014081(n) = A037800(n) + 1, n>0. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 10 2003
%e A069010 a(11)=2 since 11 is 1011 in binary with two runs of 1's. a(12)=1 since 12 is 1100 in binary with one run of 1's.
%Y A069010 Sequence in context: A029364 A122586 A079487 this_sequence A087048 A109700 A087742
%Y A069010 Adjacent sequences: A069007 A069008 A069009 this_sequence A069011 A069012 A069013
%K A069010 base,easy,nonn
%O A069010 0,6
%A A069010 Henry Bottomley (se16(AT)btinternet.com), Apr 02 2002

%I A087048
%S A087048 1,1,2,1,1,1,2,2,2,1,2,2,1,2,1,2,2,2,1,1,2,2,4,1,2,1,2,2,1,2,2,2,2,2,2,
%T A087048 1,2,2,4,1,1,2,4,2,1,2,1,1,2,4,2,1,2,2,2,2,4,1,4,2,4,3,1,2,2,4,1,4,2,1,
%U A087048 4,4,2,1,2,2,2,1,2,2,2,2,4,1,1,2,2,4,4,2,2,1,2,2,2,4,4,4,2,3,2,1,2,2,4
%N A087048 Class numbers of indefinite quadratic forms over the integers in two variables with discriminant D = D(n) = A079896(n), n>=0.
%C A087048 An indefinite quadratic form over the integers in two variables F(x,y) := a*x^2 + b*x*y + c*y^2 has discriminant D := b^2 - 4*a*c >0 not a square (a and c non-vanishing); that is D=D(n)= A079896(n) = [5,8,12,13,17,20,21,...], n>=0.
%C A087048 For a given discriminant D from A079896(n) a reduced form [a,b,c] is defined by b>0 and f(D)-min(|2*a|,|2*c|) =< b < f(D), with f(D) := ceiling(sqrt(D)).
%C A087048 For a given discriminant D from A079896(n) every primitive reduced form [a,b,c] defines a periodic chain of such forms by applying repeatedly the transformation R(t)*[a,b,c]=[a'(t),b'(t),c'(t)]=[c,-b+2*c*t,F(1,t)] with uniquely defined t= ceiling((f(D)+b)/(2*c))-1 if c>0 and t=-(ceiling((f(D)+b)/(2*|c|)-1)) if c<0. The number of such different) periodic chains of primitive reduced forms is called the class number for this (indefinite) discriminant D from A079896(n).
%C A087048 A primitive form [a,b,c] has gcd(a,b,c)=1.
%D A087048 A. Scholz and B. Schoeneberg, Einfuehrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch. 31, pp. 112 ff.
%H A087048 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Class number theory</a>
%H A087048 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/OEIS/A087048.text">Table for n=0,...,135</a>.
%e A087048 n=2, D(2) = A079896(2) = 12, a(2) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (both with period length 2): [[ -2, 2, 1], [1, 2, -2]] and [[ -1, 2, 2], [2, 2, -1]].
%e A087048 n=13, D(13) = A079896(13) = 40, a(13) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (with period length 6 resp. 2): [[ -3, 2, 3], [3, 4, -2], [ -2, 4, 3], [3, 2, -3], [ -3, 4, 2], [2, 4, -3]] and [[ -1, 6, 1], [1, 6, -1]].
%e A087048 n=35, D(35) = A079896(35) = 89, a(35) = 1 because there is only one periodic chain of primitive reduced forms [a,b,c] (with period length 14): [[ -5, 3, 4], [4, 5, -4], [ -4, 3, 5], [5, 7, -2], [ -2, 9, 1], [1, 9, -2], [ -2, 7, 5], [5, 3, -4], [ -4, 5, 4], [4, 3, -5], [ -5, 7, 2], [2, 9, -1], [ -1, 9, 2], [2, 7, -5]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the form [1, 9, -2].
%e A087048 n=62, D(62) = A079896(62) = 148, a(62) = 3 because there are three periodic chains of primitive reduced forms [a,b,c] (with period length 6 and 6 and 2, resp.): [[ -7, 6, 4], [4, 10, -3], [ -3, 8, 7], [7, 6, -4], [ -4, 10, 3], [3, 8, -7]] and [[ -4, 6, 7], [7, 8, -3], [ -3, 10, 4], [4, 6, -7], [ -7, 8, 3], [3, 10, -4]] and [[ -1, 12, 1], [1, 12, -1]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the forms [4, 10, -3] and [3, 10, -4] and [1, 12, -1], resp.
%Y A087048 See A006375 for another version. Cf. A079896.
%Y A087048 Sequence in context: A122586 A079487 A069010 this_sequence A109700 A087742 A072530
%Y A087048 Adjacent sequences: A087045 A087046 A087047 this_sequence A087049 A087050 A087051
%K A087048 nonn
%O A087048 0,3
%A A087048 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 07 2003

%I A109700
%S A109700 1,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,1,2,1,1,1,2,2,2,1,2,3,4,2,2,3,5,4,3,
%T A109700 3,6,6,6,4,6,7,9,7,7,8,11,11,11,9,12,14,16,13,14,16,21,20,19,18,24,26,
%U A109700 27,24,27,31,36,34,34,35,43,45,47,43,49,55,62,58,59,63,75,77,77,75,87
%N A109700 Number of partitions of n into parts each equal to 4 mod 5.
%F A109700 G.f.=1/product(1-x^(4+5j), j=0..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%e A109700 a(30)=2 since 30 = 14+4+4+4+4 = 9+9+4+4+4
%p A109700 g:=1/product(1-x^(4+5*j),j=0..25): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..90); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%Y A109700 Sequence in context: A079487 A069010 A087048 this_sequence A087742 A072530 A090455
%Y A109700 Adjacent sequences: A109697 A109698 A109699 this_sequence A109701 A109702 A109703
%K A109700 nonn
%O A109700 0,19
%A A109700 Erich Friedman (efriedma(AT)stetson.edu), Aug 07 2005
%E A109700 More terms from Michael Somos, Aug 10 2005

%I A087742
%S A087742 2,1,1,1,2,2,2,1,2,4,8,2,5,6,6,4,4,2,5,1,4,5,7,8,3,6,2,2,8,2,4,5,13,7,8,
%T A087742 9,13,6,15,9,12,11,8,18,6,22,26,2,24,24,8,14,32,17,31,29,21,23,13,21,18,
%U A087742 6,12,29,28,4,23,39,11,3,21,17,14,24,20,26,20,57,10,20,23,28,40,36,30
%N A087742 a(n) = 1+Abs[Prime[A005185[n]]-A005185[Prime[n]]].
%C A087742 A "commutator" between the Hofstadter A005185 sequence and the primes.
%t A087742 Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]] Hofstadter[1] = Hofstadter[2] = 1 digits=200 a=Table[1+Abs[Prime[Hofstadter[n]]-Hofstadter[Prime[n]]], {n, 1, digits}]
%Y A087742 Cf. A000040, A005185.
%Y A087742 Sequence in context: A069010 A087048 A109700 this_sequence A072530 A090455 A086288
%Y A087742 Adjacent sequences: A087739 A087740 A087741 this_sequence A087743 A087744 A087745
%K A087742 nonn
%O A087742 1,1
%A A087742 Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 01 2003
%E A087742 Edited by njas, Nov 08 2005

%I A072530
%S A072530 0,0,0,0,1,0,1,2,1,1,1,2,2,2,1,3,3,3,2,3,1,3,3,5,2,4,2,4,3,4,3,4,4,3,2,
%T A072530 6,4,5,2,6,4,6,3,6,4,5,5,7,4,6,4,5,4,8,3,5,4,7,5,9,3,7,5,8,5,7,3,8,4,8,
%U A072530 5,10,6,7,5,8,4,9,6,9,7,8,4,10,5,7,6,8,7,12,5,8,8,8,5,12,6,10,5,10,5
%N A072530 Number of primes p such that n divided by p leaves a prime remainder.
%C A072530 Is there any n > 6 such that a(n) =0 ?
%e A072530 a(17) = 3: there are 3 primes viz. 3, 5 and 7 which leave prime remainders on dividing 17.
%t A072530 Table[Count[PrimeQ[Table[Mod[w, Prime[j]], {j, 1, PrimePi[w]}]], True], {w, 1, 256}]
%Y A072530 Cf. A072531.
%Y A072530 Sequence in context: A087048 A109700 A087742 this_sequence A090455 A086288 A104360
%Y A072530 Adjacent sequences: A072527 A072528 A072529 this_sequence A072531 A072532 A072533
%K A072530 nonn
%O A072530 1,8
%A A072530 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 01 2002
%E A072530 More terms from Labos E. (labos(AT)ana.sote.hu), Aug 02 2002

%I A090455
%S A090455 0,1,0,2,1,1,1,2,2,2,2,1,0,1,1,3,3,3,0,2,0,2,0,2,0,1,1,2,1,0,2,2,1,2,1,
%T A090455 3,2,1,1,3,2,2,3,0,0,1,0,5,2,2,1,4,1,3,3,1,0,1,1,0,0,1,1,5,3,4,2,2,3,3,
%U A090455 0,4,4,3
%V A090455 0,-1,0,-2,-1,-1,1,-2,-2,-2,-2,-1,0,-1,-1,-3,-3,-3,0,-2,0,-2,0,-2,0,-1,-1,-2,-1,0,-2,
%W A090455 -2,-1,-2,-1,-3,-2,-1,-1,-3,-2,-2,-3,0,0,-1,0,-5,-2,-2,-1,-4,-1,-3,3,-1,0,-1,1,0,0,1,1,
%X A090455 -5,-3,-4,-2,-2,-3,-3,0,-4,-4,-3
%N A090455 Difference between numbers of binary 1's of n and n-th prime.
%C A090455 a(n) = A000120(n) - A014499(n);
%C A090455 a(A071600(n))=a(A049084(A072439(n)))=0; a(A049084(A090456(n)))<0; a(A049084(A090457(n)))>0.
%Y A090455 Cf. A007088, A004676, A090431.
%Y A090455 Sequence in context: A109700 A087742 A072530 this_sequence A086288 A104360 A079951
%Y A090455 Adjacent sequences: A090452 A090453 A090454 this_sequence A090456 A090457 A090458
%K A090455 sign,base
%O A090455 1,4
%A A090455 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 01 2003

%I A086288
%S A086288 0,1,1,1,1,2,1,1,1,2,2,2,2,1,2,2,2,2,1,1,2,3,1,2,2,2,3,2,2,1,2,2,2,3,2,
%T A086288 1,3,2,2,2,1,3,3,2,2,2,3,2,2,3,1,3,1,2,3,2,2,3,2,2,3,2,3,2,2,2,2,4,2,2,
%U A086288 2,3,1,2,2,3,1,3,3,2,3,3,3,2,2,3,1,3,3,2,3,2,2,2,2,4,2,2,2,3
%N A086288 Number of distinct prime factors of 7-smooth numbers.
%C A086288 a(n) = A001221(A002473(n));
%C A086288 a(n) <= 4.
%Y A086288 Cf. A086289, A086290.
%Y A086288 Sequence in context: A087742 A072530 A090455 this_sequence A104360 A079951 A095407
%Y A086288 Adjacent sequences: A086285 A086286 A086287 this_sequence A086289 A086290 A086291
%K A086288 nonn
%O A086288 1,6
%A A086288 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 15 2003

%I A104360
%S A104360 0,1,1,1,1,1,2,1,1,1,2,2,2,2,1,2,3,3,1,3,3,3,1,2,4,4,2,2,5,4,2,3,2,3,1,
%T A104360 3,4,3,5,3,4,4,5,1,3,4,2,2,2
%N A104360 Number of distinct prime factors of A104350(n) - 1.
%C A104360 a(n) = A001221(A104357(n)).
%H A104360 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a104350.txt">Products of largest prime factors of numbers <= n</a>
%Y A104360 Cf. A104368, A066877, A054989.
%Y A104360 Sequence in context: A072530 A090455 A086288 this_sequence A079951 A095407 A038605
%Y A104360 Adjacent sequences: A104357 A104358 A104359 this_sequence A104361 A104362 A104363
%K A104360 nonn
%O A104360 2,7
%A A104360 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 06 2005

%I A079951
%S A079951 0,0,1,1,1,2,1,1,1,2,2,2,2,2,1,2,1,3,2,2,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,
%T A079951 2,3,1,1,2,1,3,2,2,2,2,3,2,1,3,2,2,3,1,1,1,2,2,3,3,2,2,2,2,3,2,3,3,1,3,
%U A079951 2,1,2,3,2,1,2,3,2,3,2,4,2,2,2,2,2,3,3,3,1,1,1,2,2,1,2,3,2,3,3,2,1,2,3
%N A079951 Number of primes p with prime(n)==1 modulo 2*p.
%e A079951 n=6: prime(6)=13 and 13 mod(2*2)=1, 13 mod(2*3)=1, 13 mod(2*5)=3, 13 mod(2*7)=13, therefore a(6)=2.
%Y A079951 Cf. A079950, A079952.
%Y A079951 Sequence in context: A090455 A086288 A104360 this_sequence A095407 A038605 A113121
%Y A079951 Adjacent sequences: A079948 A079949 A079950 this_sequence A079952 A079953 A079954
%K A079951 nonn
%O A079951 1,6
%A A079951 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 19 2003

%I A095407
%S A095407 0,1,1,1,1,2,1,1,1,2,2,2,2,2,2,1,2,2,2,2,2,3,2,2,1,3,1,2,2,3,2,1,3,3,2,
%T A095407 2,2,3,3,2,2,3,2,3,2,3,2,2,1,2,3,3,2,2,3,2,3,3,2,3,2,3,2,1,3,4,2,3,3,3,
%U A095407 2,2,2,3,2,3,3,4,2,2,1,3,2,3,3,3,3,3,2,3,3,3,3,3,3,2,2,2,3,2,3,4,3,3,3
%N A095407 Total number of decimal digits of all distinct prime factors of n.
%e A095407 n=22: prime set={2,11}, a[22]=1+2=3.
%t A095407 ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] pl[x_] :=Length[tdp[x]] nl[x_] :=Length[IntegerDigits[x]] t1=Table[nl[w], {w, 1, 1000}];t2=Table[pl[w], {w, 1, 1000}];t2-t1
%Y A095407 Cf. A055642.
%Y A095407 Sequence in context: A086288 A104360 A079951 this_sequence A038605 A113121 A109036
%Y A095407 Adjacent sequences: A095404 A095405 A095406 this_sequence A095408 A095409 A095410
%K A095407 base,nonn
%O A095407 1,6
%A A095407 Labos E. (labos(AT)ana.sote.hu), Jun 21 2004

%I A038605
%S A038605 2,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,
%T A038605 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,4,
%U A038605 4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A038605 Floor( n-th prime/n ).
%F A038605 The prime number theorem is equivalent to the statement a(n) ~ log n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
%Y A038605 Cf. A000040.
%Y A038605 Sequence in context: A104360 A079951 A095407 this_sequence A113121 A109036 A085031
%Y A038605 Adjacent sequences: A038602 A038603 A038604 this_sequence A038606 A038607 A038608
%K A038605 nonn
%O A038605 1,1
%A A038605 Vasiliy Danilov (danilovv(AT)usa.net) 1998 Jul
%E A038605 Corrected by David Wasserman (wasserma(AT)spawar.navy.mil), Feb 23 2006

%I A113121
%S A113121 1,1,1,1,1,1,1,2,1,1,1,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,
%T A113121 2,2,3,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,3,2,2,
%U A113121 3,3,3,3,3,3,3,4,3,3,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,3,2,2,2,3,3,3,3
%N A113121 a(n) = smallest number of syllables necessary to communicate n by phone, in English.
%C A113121 Without "and"; 0 spelled "O", like in "Five o'clock".
%H A113121 The Math Forum, <a href="http://mathforum.org/library/drmath/view/62467.html">Ask Dr. Math</a>.
%e A113121 a(11)=2 because "one-one" is shorter than "e-le-ven";
%e A113121 a(69)=2 because "six-nine", and not "six-ty-nine";
%e A113121 a(100)=2 because "hundred" and not "one-O-O"
%Y A113121 Sequence in context: A079951 A095407 A038605 this_sequence A109036 A085031 A029342
%Y A113121 Adjacent sequences: A113118 A113119 A113120 this_sequence A113122 A113123 A113124
%K A113121 base,easy,nonn,word
%O A113121 0,8
%A A113121 Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Jan 03 2006

%I A109036
%S A109036 1,0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,2,2,2,3,2,2,2,1,2,2,3,2,3,2,3,3,2,
%T A109036 3,3,2,2,3,1,2,3,3,3,2,3,3,5,3,2,3,4,4,4,5,5,6,4,4,5,3,3,4,7,3,5,6,6,7,
%U A109036 7,7,6,6,3,5,7,8,7,8,7,9,4,5,9,5,5,6,10,4,6,9,11,11,10,10,11,8,7,6,7,7
%N A109036 Number of irreducible partitions into smaller squares. A partition is irreducible if no subpartition with 2 or more parts sums to a square smaller than n.
%C A109036 This is the same as A109035 except for the values at squares. Conjecture that lim_{n->\inf} a(n) = \inf.
%e A109036 a(10)=1 for the partition [9,1]. [4^2,1^2], [4,1^6], and [1^10] are all excluded because they contain subpartitions [4^2,1] or [1^4] summing to a square.
%Y A109036 Cf: A109035, A001156.
%Y A109036 Sequence in context: A095407 A038605 A113121 this_sequence A085031 A029342 A029408
%Y A109036 Adjacent sequences: A109033 A109034 A109035 this_sequence A109037 A109038 A109039
%K A109036 nonn
%O A109036 0,13
%A A109036 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 16 2005

%I A085031
%S A085031 1,1,1,1,2,1,1,1,2,2,2,2,2,3,2,2,4,1,2,2,1,1,4,1,3,3,2,2,1,1,2,3,2,2,3,
%T A085031 3,5,2,2,2,2,1,4,3,3,2,3,2,3,1,3,3,3,2,2,4,3,3,3,4,3,1,4,3,4,3,2,2,2,5,
%U A085031 1,3,4,3,3,2,2,4,3,3,2,3,7,2,3,1,4,2,3,1,2
%N A085031 Number of prime factors of cyclotomic(n,6), which is A019324(n), the value of the n-th cyclotomic polynomial evaluated at x=6.
%C A085031 The Mobius transform of this sequence yields A057955, number of prime factors of 6^n-1.
%D A085031 See references at A085021
%t A085031 Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 6]]][[2]], {n, 1, 100}]
%Y A085031 Cf. A019324, A057955, A085021.
%Y A085031 Sequence in context: A038605 A113121 A109036 this_sequence A029342 A029408 A029415
%Y A085031 Adjacent sequences: A085028 A085029 A085030 this_sequence A085032 A085033 A085034
%K A085031 nonn
%O A085031 1,5
%A A085031 T. D. Noe (noe(AT)sspectra.com), Jun 19 2003

%I A029342
%S A029342 1,0,0,0,1,1,0,0,1,2,1,1,1,2,2,2,2,2,3,3,4,3,4,4,5,5,5,
%T A029342 6,6,7,7,8,8,9,9,10,11,11,12,12,14,14,15,15,17,18,18,19,
%U A029342 20,22,22,23,24,26,27,28,29,30,32,33,35,35,37,39,41,42
%N A029342 Expansion of 1/((1-x^4)(1-x^5)(1-x^9)(1-x^11)).
%Y A029342 Sequence in context: A113121 A109036 A085031 this_sequence A029408 A029415 A128494
%Y A029342 Adjacent sequences: A029339 A029340 A029341 this_sequence A029343 A029344 A029345
%K A029342 nonn
%O A029342 0,10
%A A029342 njas

%I A029408
%S A029408 1,0,0,0,0,1,0,0,1,0,1,1,1,1,0,1,2,1,1,1,2,2,2,2,3,2,2,
%T A029408 3,3,3,3,3,5,4,4,5,5,5,5,5,7,6,6,7,8,8,8,8,10,9,9,10,11,
%U A029408 11,11,12,14,13,13,14,16,15,15,16,18,18,18,19,21,20,21
%N A029408 Expansion of 1/((1-x^5)(1-x^8)(1-x^11)(1-x^12)).
%Y A029408 Sequence in context: A109036 A085031 A029342 this_sequence A029415 A128494 A110730
%Y A029408 Adjacent sequences: A029405 A029406 A029407 this_sequence A029409 A029410 A029411
%K A029408 nonn
%O A029408 0,17
%A A029408 njas

%I A029415
%S A029415 1,0,0,0,0,0,1,1,1,0,0,1,1,1,2,1,1,1,2,2,2,2,3,2,3,3,3,
%T A029415 3,4,4,5,4,5,5,5,6,7,6,7,7,8,8,9,9,10,9,11,11,12,12,13,
%U A029415 13,14,14,16,16,17,17,18,18,20,20,22,22,23,23,25,25,27
%N A029415 Expansion of 1/((1-x^6)(1-x^7)(1-x^8)(1-x^11)).
%Y A029415 Sequence in context: A085031 A029342 A029408 this_sequence A128494 A110730 A050431
%Y A029415 Adjacent sequences: A029412 A029413 A029414 this_sequence A029416 A029417 A029418
%K A029415 nonn
%O A029415 0,15
%A A029415 njas

%I A128494
%S A128494 1,1,1,0,1,1,0,1,1,1,1,1,2,1,1,1,2,2,3,1,1,0,2,4,3,4,1,1,0,2,4,7,4,5,1,
%T A128494 1,1,2,6,7,11,5,6,1,1,1,3,6,13,11,16,6,7,1,1,0,3,9,13,24,16,22,7,8,1,1,
%U A128494 0,3,9,22,24,40,22,29,8,9,1,1,1,3,12,22,46,40,62,29,37,9,10,1,1,1,4,12
%V A128494 1,1,1,0,1,1,0,-1,1,1,1,-1,-2,1,1,1,2,-2,-3,1,1,0,2,4,-3,-4,1,1,0,-2,4,7,-4,-5,1,1,1,
%W A128494 -2,-6,7,11,-5,-6,1,1,1,3,-6,-13,11,16,-6,-7,1,1,0,3,9,-13,-24,16,22,-7,-8,1,1,0,-3,9,
%X A128494 22,-24,-40,22,29,-8,-9,1,1,1,-3,-12,22,46,-40,-62,29,37,-9,-10,1,1,1,4,-12
%N A128494 Coefficient table for sums of Chebyshev's S-Polynomials.
%C A128494 See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.
%C A128494 This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k)=a(2*p+1,2*k)=A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1)=a(2*p+2,2*k+1)=A059260(p+k+1,2*k+1)*(-1)^(p+k), k>=0.
%H A128494 W. Lang,<a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A128494.text">First 15 rows.</a>
%F A128494 S(1;n,x):=sum(S(k,x),k=0..n)=sum(a(n,m)*x^m,m=0..n), n>=0.
%F A128494 a(n,m)=[x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
%F A128494 G.f. for column sequence nr. m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
%e A128494 [1]; [1,1]; [0,1,1]; [0,-1,1,1]; [1,-1,-2,1,1]; [1,2,-2,-3,1,1]; ...
%e A128494 Row polynomial S(1;4,x)=1-x-2*x^2+x^3+x^4 = sum(S(k,x),k=0..4).
%e A128494 S(4,y)*S(5,y)/y=3-13*y^2+16*y^4-7*y^6+y^8, with y=sqrt(2+x) this becomes S(1;4,x).
%Y A128494 Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
%Y A128494 Cf. A128495 for S(2; n, x) coefficient table.
%Y A128494 The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
%Y A128494 For m>=1 the column sequences (without leading zeros) are of the form a(m, 2*k)=a(m, 2*k+1)=((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.
%Y A128494 Sequence in context: A029342 A029408 A029415 this_sequence A110730 A050431 A051574
%Y A128494 Adjacent sequences: A128491 A128492 A128493 this_sequence A128495 A128496 A128497
%K A128494 sign,tabl,easy
%O A128494 0,13
%A A128494 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Apr 04 2007

%I A110730
%S A110730 1,1,1,2,1,1,1,2,2,3,1,1,1,1,2,2,2,3,3,4,1,1,1,1,1,2,2,2,2,3,3,3,4,4,5,
%T A110730 1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,1,1,1,1,1,1,1,2,2,2,2,2,2,3,
%U A110730 3,3,3,3,4,4,4,4,5,5,5,6,6,7,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3
%N A110730 n ones followed by (n-1) 2s followed by (n-3) 3s ...finally one n.
%Y A110730 Cf. A004736.
%Y A110730 Sequence in context: A029408 A029415 A128494 this_sequence A050431 A051574 A029386
%Y A110730 Adjacent sequences: A110727 A110728 A110729 this_sequence A110731 A110732 A110733
%K A110730 base,easy,nonn
%O A110730 1,4
%A A110730 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 09 2005

%I A050431
%S A050431 1,1,1,2,1,1,1,2,2,3,1,2,3,2,1,3,2,2,1,3,1,2,3,2,2,3,3,4,2,3,3,3,3,1,2,
%T A050431 2,3,2,3,4,3,2,3,2,2,1,3,3,3,3,2,4,3,3,2,4,3,2,1,3,3,3,2,3,1,2,3,4,3,3,
%U A050431 3,2,2,3,2,2,3,3,3,4,4,5,3,4,4,4,3,2,2,3
%N A050431 Length of longest palindromic subword of (n base 3).
%Y A050431 Sequence in context: A029415 A128494 A110730 this_sequence A051574 A029386 A060502
%Y A050431 Adjacent sequences: A050428 A050429 A050430 this_sequence A050432 A050433 A050434
%K A050431 nonn,base
%O A050431 1,4
%A A050431 Clark Kimberling (ck6(AT)evansville.edu)

%I A051574
%S A051574 1,1,1,1,2,1,1,1,2,2,3,1,3,1,1,1,3,1,5,3,2,1,1,1,2,2,2,4,7,1,2,2,5,6,3,
%T A051574 1,6,3,2,1,4,1,2,1,2,2,2,1,4,5,9,5,8,4,7,3,9,6,8,2,6,3,1,4,11,5,9,4,5,
%U A051574 1,4,1,7,4,2,4,8,3,4,1,6,11,15,3,7,5,4,5,9,1,5,3,2,2,1,1,4,2,7,7,19,8
%N A051574 a(n) = number of k, 1<=k<=n, such that n*k) divides binomial(n,k).
%e A051574 a(11)=3 since k=1, k=3, k=6 are only solutions to 11*k divides binomial(11,k).
%Y A051574 Sequence in context: A128494 A110730 A050431 this_sequence A029386 A060502 A035439
%Y A051574 Adjacent sequences: A051571 A051572 A051573 this_sequence A051575 A051576 A051577
%K A051574 nonn
%O A051574 1,5
%A A051574 Leroy Quet (qq-quet(AT)mindspring.com)

%I A029386
%S A029386 1,0,0,0,0,1,1,0,1,0,1,1,2,1,1,1,2,2,3,1,3,2,3,3,5,3,4,
%T A029386 3,5,5,7,4,7,5,7,7,10,7,9,7,11,10,13,9,13,11,14,13,18,13,
%U A029386 17,14,19,18,22,17,23,19,24,22,29,23,28,24,31,29,35,28
%N A029386 Expansion of 1/((1-x^5)(1-x^6)(1-x^8)(1-x^12)).
%Y A029386 Sequence in context: A110730 A050431 A051574 this_sequence A060502 A035439 A059111
%Y A029386 Adjacent sequences: A029383 A029384 A029385 this_sequence A029387 A029388 A029389
%K A029386 nonn
%O A029386 0,13
%A A029386 njas

%I A060502
%S A060502 0,1,1,2,1,1,1,2,2,3,2,2,1,2,1,2,2,2,1,1,2,2,1,1,1,2,2,3,2,2,2,3,3,4,3,
%T A060502 3,2,3,2,3,3,3,2,2,3,3,2,2,1,2,2,3,2,2,1,2,2,3,2,2,2,3,2,3,3,3,2,2,3,3,
%U A060502 2,2,1,2,1,2,2,2,2,3,2,3,3,3,1,2,1,2,2,2,2,2,2,2,2,2,1,1,2,2,1,1,2,2,3
%N A060502 Average of digits of each term in A060498, number of balls in each such siteswap juggling pattern.
%F A060502 a(n) = avg(Perm2SiteSwap3(PermUnrank3R(n)))
%p A060502 Perm2SiteSwap3 := proc(p) local ip,n,i,a; n := nops(p); ip := convert(invperm(convert(p,'disjcyc')),'permlist',n); a := []; for i from 1 to n do if(0 = ((ip[i]-i) mod n)) then a := [op(a),0]; else a := [op(a), n-((ip[i]-i) mod n)]; fi; od; RETURN(a); end;
%Y A060502 A060500 gives average.
%Y A060502 Sequence in context: A050431 A051574 A029386 this_sequence A035439 A059111 A103502
%Y A060502 Adjacent sequences: A060499 A060500 A060501 this_sequence A060503 A060504 A060505
%K A060502 nonn
%O A060502 0,4
%A A060502 Antti Karttunen Mar 22 2001

%I A035439
%S A035439 0,0,0,1,0,1,0,1,0,1,1,2,1,1,1,2,2,3,2,3,2,4,3,6,4,6,4,7,6,9,8,10,9,11,
%T A035439 11,15,14,17,16,20,18,24,23,29,27,32,31,38,38,45,45,52,51,60,60,71,71,
%U A035439 82,81,94,94,108,111,126,128,143,146,164,169,190,195,218,221,246
%N A035439 Number of partitions of n into parts 7k+4 or 7k+6.
%Y A035439 Sequence in context: A051574 A029386 A060502 this_sequence A059111 A103502 A069545
%Y A035439 Adjacent sequences: A035436 A035437 A035438 this_sequence A035440 A035441 A035442
%K A035439 nonn
%O A035439 1,12
%A A035439 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A059111
%S A059111 2,1,1,1,2,2,3,2,3,5,4,7,7,6,6,8,10,8,11,11,9,10,10,12,16,16,14,13,11,
%T A059111 10,20,20,21,19,24,21,23,24,24,25,26,24,29,26,25,22,30,37,36,33,32,33,
%U A059111 30,35,36,37,38,35,36,35,32,37,45,44,41,40,49,50,54,51,50,51,53,54,55
%N A059111 Ceiling(prime(n) - n*log(n)).
%H A059111 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml">How Many Primes Are There?</a>
%Y A059111 Cf. A064658, A064659, A059112.
%Y A059111 Sequence in context: A029386 A060502 A035439 this_sequence A103502 A069545 A122520
%Y A059111 Adjacent sequences: A059108 A059109 A059110 this_sequence A059112 A059113 A059114
%K A059111 nonn
%O A059111 1,1
%A A059111 Henry Bottomley (se16(AT)btinternet.com), Jan 04 2001

%I A103502
%S A103502 0,0,0,0,1,0,1,1,1,0,1,1,2,1,1,1,2,2,3,3,2,2,3,2,3,2,3,2,4,3,4,5,4,4,3,
%T A103502 3,4,4,4,3,5,3,5,5,4,4,6,5,6,6,5,5,7,6,6,6,5,5,7,5,7,7,6,7,7,6,7,7,7,6,
%U A103502 7,7,9,8,8,8,8,7,8,8,9,9,10,9,9,9,8,8,10,9,9,8,8,8,8,8,9,9,9,9,11,9,11
%N A103502 Floor of Sum_{p prime <= n} (fractional part of n/p).
%e A103502 a(6)=[{6/2}+{6/3}+{6/5}]=[0+0+1/5]=0.
%e A103502 a(6)=[{6/2}+{6/3}+{6/5}]=[0+0+1/5]=0.
%t A103502 f[n_] := Floor[Plus @@ FractionalPart[n / Table[Prime[i], {i, PrimePi[ n]}]]]; Table[ f[n], {n, 103}]
%Y A103502 Sequence in context: A060502 A035439 A059111 this_sequence A069545 A122520 A058393
%Y A103502 Adjacent sequences: A103499 A103500 A103501 this_sequence A103503 A103504 A103505
%K A103502 nonn
%O A103502 0,13
%A A103502 Carlos Alves (cjsalves(AT)gmail.com), Feb 08 2005
%E A103502 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 11 2005

%I A069545
%S A069545 1,2,1,1,1,2,2,3,3,4,2,1,3,6,4,1,3,5,1,2,1,1,1,2,5,1,1,1,1,1,2,3,1,4,1,
%T A069545 2,1,3,2,1,5,1,2,1,4,2,2
%N A069545 Liouville clusters: the number of successive occurrences of signs in Liouville function lambda(k); a(2n-1) is number of successive positive signs, while a(2n) is number of successive negative signs.
%C A069545 Related open questions. What is the limit of ratio: a(n)/n, as n->infinity? What is frequency distribution of integer k in the sequence; a(n)=k for what set of n?
%D A069545 H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
%D A069545 H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
%D A069545 R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
%H A069545 MathWorld, <a href="http://www.mathworld.wolfram.com/LiouvilleFunction.html">Liouville function</a>
%F A069545 Related to summatory Liouville function (A002819): L(m)=sum_{k=1, n} (-1)^(k-1)*a(k) where m=sum_{k=1, n} a(k).
%e A069545 a(6)=2 because the 6-th Liouville cluster consists of 2 successive negative signs: lambda(7)=lambda(8)=(-1)
%Y A069545 Cf. A008836, A002819, A001222, A028260, A026424.
%Y A069545 Sequence in context: A035439 A059111 A103502 this_sequence A122520 A058393 A131256
%Y A069545 Adjacent sequences: A069542 A069543 A069544 this_sequence A069546 A069547 A069548
%K A069545 easy,nice,nonn
%O A069545 1,2
%A A069545 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 17 2002

%I A122520
%S A122520 1,1,1,1,1,2,1,1,1,2,2,4,1,2,5,4,7,5,8,1,16,2,20,2,25,13,38,20,43,40,50,
%T A122520 71,61,103,53,161,40,235,11,317,68,436,184,563,374,685,688,815,1121,874
%V A122520 1,1,1,1,1,-2,1,1,1,-2,-2,4,1,-2,-5,4,7,-5,-8,1,16,-2,-20,-2,25,13,-38,-20,43,40,-50,
%W A122520 -71,61,103,-53,-161,40,235,-11,-317,-68,436,184,-563,-374,685,688,-815,-1121,874
%N A122520 5 X 5 vector matrix Markov: characteristic polynomial: (-1 - x - x^2 - x^3 + x^5).
%C A122520 Root sum is zero: Table[x /. NSolve[Det[M*x - IdentityMatrix[5]] == 0, x][[n]], {n, 1, 5}] Apply[Plus, %]=0
%F A122520 M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 0, -1, -1, -1}}; w[1] = {1, 1, 1, 1, 1}; w[n_] := w[n] = M.w[n - 1] a(n) = v[n][[1]]
%t A122520 M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 0, -1, -1, -1}}; w[1] = {1, 1, 1, 1, 1}; w[n_] := w[n] = M.w[n - 1] a = Table[w[n][[1]], {n, 1, 30}]
%Y A122520 Sequence in context: A059111 A103502 A069545 this_sequence A058393 A131256 A122945
%Y A122520 Adjacent sequences: A122517 A122518 A122519 this_sequence A122521 A122522 A122523
%K A122520 sign,uned
%O A122520 1,6
%A A122520 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2006

%I A058393
%S A058393 1,0,1,1,1,1,0,1,2,1,1,1,2,3,1,0,1,2,4,4,1,1,1,2,4,7,5,1,0,1,2,4,8,11,
%T A058393 6,1,1,1,2,4,8,15,16,7,1,0,1,2,4,8,16,26,22,8,1,1,1,2,4,8,16,31,42,29,
%U A058393 9,1,0,1,2,4,8,16,32,57,64,37,10,1,1,1,2,4,8,16,32,63,99,93,46,11,1,0
%N A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.
%C A058393 Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).
%F A058393 T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).
%e A058393 Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.
%Y A058393 Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.
%Y A058393 Sequence in context: A103502 A069545 A122520 this_sequence A131256 A122945 A119338
%Y A058393 Adjacent sequences: A058390 A058391 A058392 this_sequence A058394 A058395 A058396
%K A058393 nonn,tabl
%O A058393 0,9
%A A058393 Henry Bottomley (se16(AT)btinternet.com), Nov 24 2000

%I A131256
%S A131256 1,0,1,1,1,1,0,2,1,1,1,2,3,1,1,0,3,4,3,1,1,1,3,7,5,3,1,1,0,4,9,10,5,3,1,
%T A131256 11,4,13,16,11,5,3,1,1,0,5,16,26,20,11,5,3,1,1
%N A131256 A000012(signed) * A052509.
%C A131256 Row sums = A052952: (1, 1, 3, 4, 8, 12, 21, 33,...). A131257 = A052509 * A000012(signed).
%F A131256 A000012(signed) * A052509, where the signed version of A000012 = (1; -1,1; 1,-1,1;...).
%e A131256 First few rows of the triangle are:
%e A131256 1;
%e A131256 0, 1;
%e A131256 1, 1, 1;
%e A131256 0, 2, 1, 1;
%e A131256 1, 2, 3, 1, 1;
%e A131256 0, 3, 4, 3, 1, 1;
%e A131256 1, 3, 7, 5, 3, 1, 1;
%e A131256 ...
%Y A131256 Cf. A052509, A000012, A131257.
%Y A131256 Sequence in context: A069545 A122520 A058393 this_sequence A122945 A119338 A054124
%Y A131256 Adjacent sequences: A131253 A131254 A131255 this_sequence A131257 A131258 A131259
%K A131256 nonn,tabl
%O A131256 0,8
%A A131256 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 23 2007

%I A122945
%S A122945 1,1,1,1,1,1,1,1,0,1,1,1,1,2,1,1,1,2,3,1,1,1,1,3,4,0,3,1,1,1,4,5,2,6,2,
%T A122945 1,1,1,5,6,5,10,2,4,1,1,1,6,7,9,15,0,10,3,1,1,1,7,8,14,21,5,20,5,5,1,1,
%U A122945 1,8,9,20,28,14,35,5,15,4,1,1,1,9,10,27,36,28,56,0,35,9,6,1,1,1,10,11
%V A122945 1,1,-1,-1,1,1,1,-1,0,-1,-1,1,1,-2,1,1,-1,-2,3,-1,-1,-1,1,3,-4,0,3,1,1,-1,-4,5,2,-6,2,
%W A122945 -1,-1,1,5,-6,-5,10,-2,-4,1,1,-1,-6,7,9,-15,0,10,-3,-1,-1,1,7,-8,-14,21,5,-20,5,5,1,1,
%X A122945 -1,-8,9,20,-28,-14,35,-5,-15,4,-1,-1,1,9,-10,-27,36,28,-56,0,35,-9,-6,1,1,-1,-10,11
%N A122945 Recursive polynomias (p(k, x) = p(k - 1, x) - x^2*p(k - 2, x) ) used to produce a set of matrices a(i,j) at level n that then produce the characteristic polynomials which provide the triangular sequence t(n,m).
%C A122945 It was a real problem getting the matrices to agree with the polynomials: I was getting shift function polynomials!) 1 X 1 {{1}} 2 X 2 {{0, 1}, {1, -1}} 3 X 3 {{0, 1, 0}, {0, 0, 1}, {1, -1, 0}} 4 X 4 {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, -1, -1, 2}} 5 X 5 {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, -1, -2, 3, -1}}
%F A122945 p(k, x) = p(k - 1, x) - x^2*p(k - 2, x) p(k,n)->t0(i,j) t0(i,j)->a[i,j) a(i,j)->p'(n,x) p'(n,k)->t(n,m)
%e A122945 Input triangular sequence from the recurvise polynomials:
%e A122945 {{1},
%e A122945 {-1,1},
%e A122945 {-1, 1, -1},
%e A122945 {-1, 1, 0, -1},
%e A122945 {-1, 1, 1, -2, 1},
%e A122945 {-1, 1, 2, -3, 1, 1}
%e A122945 Output triangular sequence from characteristic polynomials of matrices:
%e A122945 {1},
%e A122945 {1, -1},
%e A122945 {-1, 1, 1},
%e A122945 {1, -1, 0, -1},
%e A122945 {-1, 1, 1, -2, 1},
%e A122945 {1, -1, -2, 3, -1, -1}
%t A122945 p[0, x] = 1; p[1, x] = x - 1; p[k_, x_] := p[k, x] = p[k - 1, x] - x^2*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 20}] ; An[d_] := Table[If[n == d, -w[[n]][[m]], If[m == n, 1, 0]], {n, 2, d}, {m, 1, d - 1}]; Table[An[d], {d, 2, 19}] b = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], \ {d, 2, 19}]]; Flatten[%]
%Y A122945 Sequence in context: A122520 A058393 A131256 this_sequence A119338 A054124 A096670
%Y A122945 Adjacent sequences: A122942 A122943 A122944 this_sequence A122946 A122947 A122948
%K A122945 tabl,uned,sign
%O A122945 1,14
%A A122945 Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 24 2006

%I A119338
%S A119338 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,4,4,1,1,1,2,4,6,6,1,1,1,2,4,7,11,8,
%T A119338 1,1,1,2,4,7,13,19,12,1,1,1,2,4,7,14,25,33,16,1,1,1,2,4,7,14,27,49,55,
%U A119338 22,1,1,1,2,4,7,14,28,55,93,95,29,1,1,1,2,4,7,14,28,57,111,181,158,40,1
%N A119338 Table by anti-diagonals: a(m,n) is the number of m-dimensional partitions of n up to conjugacy, for m >= 0, n >= 1.
%C A119338 Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.
%e A119338 Table starts:
%e A119338 1, 1, 1, 1, 1, 1, ...
%e A119338 1, 1, 2, 3, 4, 6, ...
%e A119338 1, 1, 2, 4, 6, 11, ...
%e A119338 1, 1, 2, 4, 7, 13, ...
%e A119338 1, 1, 2, 4, 7, 14, ...
%e A119338 ...
%Y A119338 Rows: A000012, A046682, A000786, A119266, A119267, A119340, A119341, A119342 stabilize to A119268. Transposed table is A119269. Cf. A119339, A119270, A118364, A118365.
%Y A119338 Sequence in context: A058393 A131256 A122945 this_sequence A054124 A096670 A130461
%Y A119338 Adjacent sequences: A119335 A119336 A119337 this_sequence A119339 A119340 A119341
%K A119338 nonn,tabl
%O A119338 1,9
%A A119338 Max Alekseyev (maxal(AT)cs.ucsd.edu), May 15 2006

%I A054124
%S A054124 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,4,4,1,1,1,2,4,7,5,1,1,1,2,4,8,11,
%T A054124 6,1,1,1,2,4,8,15,16,7,1,1,1,2,4,8,16,26,22,8,1,1,1,2,4,8,16,31,42,29,
%U A054124 9,1,1,1,2,4,8,16,32,57,64,37,10,1,1,1,2
%N A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.
%C A054124 Reflection of array in A054123 about vertical central line.
%H A054124 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pas.html#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A054124 T(n, 0)=T(n, n)=1 for n >= 0; T(n, k)=T(n-1, k-1)+T(n-2, k-1) for k=1, 2, ..., n-1, n >= 2.
%e A054124 Rows: {1}, {1,1}, {1,1,1}, {1,1,2,1}, {1,1,2,3,1}, ...
%Y A054124 Row sums: 1, 2, 3, 5, 8, 13, ... (Fibonacci numbers, A000045). Central numbers: 1, 1, 2, 4, 8, ... (binary powers, A000079).
%Y A054124 First n numbers of n-th column for n >= 1 form the array in A008949.
%Y A054124 Sequence in context: A131256 A122945 A119338 this_sequence A096670 A130461 A130777
%Y A054124 Adjacent sequences: A054121 A054122 A054123 this_sequence A054125 A054126 A054127
%K A054124 nonn,tabl,eigen,nice
%O A054124 0,9
%A A054124 Clark Kimberling (ck6(AT)evansville.edu)

%I A096670
%S A096670 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,5,5,1,1,1,2,5,9,8,1,1,1,2,5,12,18,
%T A096670 13,1,1,1,2,5,12,24,37,21,1,1,1,2,5,12,29,52,73,34,1,1,1,2,5,12,29,62,
%U A096670 115,146,55,1,1,1,2,5,12,29,70,140,251,293,89,1,1,1,2,5,12,29,70,156
%N A096670 Rectangular array T(n,k) read by antidiagonals; generating function of column n is 1/F(n,x), where F(n,x) is the polynomial 1 - x - x^2 - 2*x^3 -...- F(n+1)*x^n, and F(n+1) is the (n+1)st Fibonacci number, for n=0,1,2,...
%C A096670 Transpose of the array in A096669.
%e A096670 Rows
%e A096670 1 1 1 1 1
%e A096670 1 1 1 1 1
%e A096670 1 2 2 2 2
%e A096670 1 3 5 5 5
%e A096670 1 5 9 12 12
%e A096670 Column 0 has g.f. 1/(1-x)
%e A096670 Column 1 has g.f. 1/(1-x-x^2)
%e A096670 Column 2 has g.g. 1/(1-x-x^2-2*x^3).
%Y A096670 Cf. A000045, A096669, A000129.
%Y A096670 Sequence in context: A122945 A119338 A054124 this_sequence A130461 A130777 A046854
%Y A096670 Adjacent sequences: A096667 A096668 A096669 this_sequence A096671 A096672 A096673
%K A096670 nonn,tabl
%O A096670 1,9
%A A096670 Clark Kimberling (ck6(AT)evansville.edu), Jul 03 2004

%I A130461
%S A130461 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,6,4,1,1,1,1,6,12,5,1,1,1,2,6,24,20,
%T A130461 6,1,1,1,1,6,24,60,30,7,1,1,1,2,6,24,120,120,42,8,1,1,1,2,6,24,120,360,
%U A130461 210,56,9,1,1,1,2,6,24,120,720,840,336,72,10,1,1,1,2,6,24,120,720,2520
%N A130461 Triangle, antidiagonals of an array generated from A130460.
%C A130461 Rows tend to the factorials: (1, 1, 2, 6, 24,...). Row sums = A130476: (1, 2, 3, 5, 8, 15, 28, 61, 132,...).
%F A130461 Let A130460 = M, an infinite lower triangular matrix, and V = [1, 1, 1,...], the first row of an array. Peform M * V = second row,...; (n+1)-th row = M * n-th row. The triangle = antidiagonals of the array.
%e A130461 The array =
%e A130461 1,...1,...1,...1,....1,....1,...
%e A130461 1,...1,...2,...3,....4,....5,...
%e A130461 1,...1,...2,...6,...12,...20,...
%e A130461 1,...1,...2,...6,...24,...60,...
%e A130461 1,...1,...2,...6,...24,..120,...
%e A130461 1,...1,...2,...6,...24,..120,...
%e A130461 ...
%e A130461 First few rows of the triangle are:
%e A130461 1;
%e A130461 1, 1;
%e A130461 1, 1, 1;
%e A130461 1, 1, 2, 1;
%e A130461 1, 1, 2, 3, 1;
%e A130461 1, 1, 2, 6, 4, 1;
%e A130461 1, 1, 2, 6, 12, 5, 1;
%e A130461 1, 1, 2, 6, 24, 20, 6, 1;
%e A130461 1, 1, 2, 6, 24, 60, 30, 7, 1;
%e A130461 ...
%Y A130461 Cf. A130460, A130476, A130477, A130478.
%Y A130461 Sequence in context: A119338 A054124 A096670 this_sequence A130777 A046854 A066170
%Y A130461 Adjacent sequences: A130458 A130459 A130460 this_sequence A130462 A130463 A130464
%K A130461 nonn,tabl
%O A130461 0,9
%A A130461 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 28 2007

%I A130777
%S A130777 1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,3,3,4,1,1,1,3,6,4,5,1,1,1,4,6,10,5,6,1,
%T A130777 1,1,4,10,10,15,6,7,1,1,1,5,10,20,15,21,7,8,1,1,1,5,15,20,35,21,28,8,9,
%U A130777 1,1,1,6,15,35,35,56,28,36,9,10,1,1
%V A130777 1,-1,1,-1,-1,1,1,-2,-1,1,1,2,-3,-1,1,-1,3,3,-4,-1,1,-1,-3,6,4,-5,-1,1,1,-4,-6,10,5,-6,
%W A130777 -1,1,1,4,-10,-10,15,6,-7,-1,1,-1,5,10,-20,-15,21,7,-8,-1,1,-1,-5,15,20,-35,-21,28,8,
%X A130777 -9,-1,1,1,-6,-15,35,35,-56,-28,36,9,-10,-1,1
%N A130777 Inverse of triangle in A061554.
%C A130777 Signed version of A046854.
%e A130777 Triangle begins:
%e A130777 1;
%e A130777 -1, 1;
%e A130777 -1, -1, 1;
%e A130777 1, -2, -1, 1;
%e A130777 1, 2, -3, -1, 1;
%e A130777 -1, 3, 3, -4, -1, 1;
%e A130777 -1, -3, 6, 4, -5, -1, 1;
%e A130777 1, -4, -6, 10, 5, -6, -1, 1;
%e A130777 1, 4, -10, -10, 15, 6, -7, -1, 1 ;...
%Y A130777 Cf. A066170 A046854.
%Y A130777 Sequence in context: A054124 A096670 A130461 this_sequence A046854 A066170 A071773
%Y A130777 Adjacent sequences: A130774 A130775 A130776 this_sequence A130778 A130779 A130780
%K A130777 sign,tabl
%O A130777 0,8
%A A130777 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 14 2007

%I A046854
%S A046854 1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,3,3,4,1,1,1,3,6,4,5,1,1,1,4,6,10,5,
%T A046854 6,1,1,1,4,10,10,15,6,7,1,1,1,5,10,20,15,21,7,8,1,1,1,5,15,20,35,21,
%U A046854 28,8,9,1,1,1,6,15,35,35,56,28,36,9,10,1,1,1,6,21,35,70,56,84,36,45
%N A046854 Triangle in which k-th entry of row n is binomial[ Floor[n/2 + k/2], k].
%C A046854 Row sums are F(n+2). Diagonal sums are A016116. - Paul Barry (pbarry(AT)wit.ie), Jul 07 2004
%C A046854 Riordan array (1/(1-x), x/(1-x^2)). Matrix inverse is A106180. - Paul Barry (pbarry(AT)wit.ie), Apr 24 2005
%F A046854 G.f.: (1+x) / (1-xy-x^2). - Ralf Stephan, Feb 13 2005
%F A046854 Triangle = A097806 * A049310, as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 28 2007
%e A046854 {1}, {1, 1}, {1, 1, 1}, {1, 2, 1, 1}, {1, 2, 3, 1, 1}, ...
%t A046854 Table[ Binomial[ Floor[ n/2 +k/2 ], k ], {n, 0, 16}, {k, 0, n} ]
%Y A046854 Reflected version of A065941, which is considered the main entry. Probably a signed version is A066170. A deficient version is in A030111.
%Y A046854 Cf. A066170.
%Y A046854 Cf. A097806, A049310.
%Y A046854 Sequence in context: A096670 A130461 A130777 this_sequence A066170 A071773 A000188
%Y A046854 Adjacent sequences: A046851 A046852 A046853 this_sequence A046855 A046856 A046857
%K A046854 nonn,tabl,easy
%O A046854 0,8
%A A046854 Wouter Meeussen (wouter.meeussen(AT)pandora.be)

%I A066170
%S A066170 1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,3,3,4,1,1,1,3,6,4,5,1,1,1,4,6,10,5,6,
%T A066170 1,1,1,4,10,10,15,6,7,1,1,1,5,10,20,15,21,7,8,1,1,1,5,15,20,35,21,28,8,
%U A066170 9,1,1,1,6,15,35,35,56,28,36,9,10,1,1,1,6,21,35,70,56,84,36,45,10,11,1
%V A066170 1,-1,1,1,-1,-1,-1,2,1,-1,1,-2,-3,1,1,-1,3,3,-4,-1,1,1,-3,-6,4,5,-1,-1,-1,4,6,-10,-5,
%W A066170 6,1,-1,1,-4,-10,10,15,-6,-7,1,1,-1,5,10,-20,-15,21,7,-8,-1,1,1,-5,-15,20,35,-21,-28,
%X A066170 8,9,-1,-1,-1,6,15,-35,-35,56,28,-36,-9,10,1,-1,1,-6,-21,35,70,-56,-84,36,45,-10,-11
%N A066170 Triangle giving coefficients of characteristic function of n X n matrix in which the left upper half and the antidiagonal are filled with 1's, and the right lower half is filled with 0's.
%C A066170 The table begins {1}; {-1, 1}; {1, -1, -1}; {-1, 2, 1, -1}; ...
%D A066170 J. R. Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330. [See p. 328.]
%D A066170 Henry W. Gould, "A Variant of Pascal's Triangle", The Fibonacci Quarterly,3;4 Dec. 1965, pp. 257-271.
%D A066170 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)
%D A066170 P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
%F A066170 Polynomial recursion: p[k, x] = x*p[k - 1, x] - p[k - 2, x]. Also T(n, k) = (-1)^Floor[(k + 1)/2]*binomial[n - Floor[(k + 1)/2], Floor[k/2]] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
%e A066170 The characteristic function of
%e A066170 ( 1 1 1 )
%e A066170 ( 1 1 0 )
%e A066170 ( 1 0 0 )
%e A066170 is f(x) = -x^3 + 2x^2 + x - 1, so the 3rd row is {-1; 2; 1; -1}.
%e A066170 Triangle (with rows reversed) begins:
%e A066170 1
%e A066170 1, -1
%e A066170 1, -1, -1,
%e A066170 1, -1, -2, 1
%e A066170 1, 1, -3, 2, 1
%e A066170 1, -1, -4, 3, 3,-1
%e A066170 1, -1, -5, 4, 6, -3, -1
%e A066170 1, -1, -6, 5, 10,-6, -4, 1
%e A066170 1, -1, -7, 6, 15, -10, -10, 4, 1
%e A066170 1, -1, -8, 7, 21, -15, -20, 10, 5,-1
%e A066170 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1
%t A066170 (* Triangular*) T[n_, k_] := (-1)^Floor[(k + 1)/2]*Binomial[n - Floor[(k + 1)/2], Floor[k/2]] Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
%t A066170 (* Polynomial recursion*) p[0, x] = 1; p[1, x] = x - 1; p[2, x] = x^2 - x - 1; p[3, x] = x^3 - x^2 - 2*x + 1; p[k_, x_] := p[k, x] = x*p[k - 1, x] - p[k - 2, x] ; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
%Y A066170 Signed version of A046854.
%Y A066170 Cf. A007700, A059455, A065941. For another version see A030111.
%Y A066170 Sequence in context: A130461 A130777 A046854 this_sequence A071773 A000188 A097886
%Y A066170 Adjacent sequences: A066167 A066168 A066169 this_sequence A066171 A066172 A066173
%K A066170 sign,easy,tabl
%O A066170 0,8
%A A066170 Floor van Lamoen (fvlamoen(AT)hotmail.com), Dec 14 2001
%E A066170 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 02 2002

%I A071773
%S A071773 1,1,1,2,1,1,1,2,3,1,1,2,1,1,1,2,1,3,1,2,1,1,1,2,5,1,3,2,1,1,1,2,1,1,1,
%T A071773 6,1,1,1,2,1,1,1,2,3,1,1,2,7,5,1,2,1,3,1,2,1,1,1,2,1,1,3,2,1,1,1,2,1,1,
%U A071773 1,6,1,1,5,2,1,1,1,2,3,1,1,2,1,1,1,2,1,3,1,2,1,1,1,2,1,7,3,10,1,1,1,2
%N A071773 GCD(sfk(n),n/sfk(n)), where sfk(n)=A007947(n) is the square-free kernel of n.
%C A071773 a(n)=gcd(A007947(n), A003557(n));
%C A071773 n is square-free iff a(n)=1.
%H A071773 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree.</a>
%F A071773 Multiplicative with p^e -> p^ceil((e-1)/e), p prime.
%Y A071773 Cf. A005117.
%Y A071773 Sequence in context: A130777 A046854 A066170 this_sequence A000188 A097886 A088863
%Y A071773 Adjacent sequences: A071770 A071771 A071772 this_sequence A071774 A071775 A071776
%K A071773 nonn,mult
%O A071773 1,4
%A A071773 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 24 2002

%I A000188
%S A000188 1,1,1,2,1,1,1,2,3,1,1,2,1,1,1,4,1,3,1,2,1,1,1,2,5,1,3,2,1,1,
%T A000188 1,4,1,1,1,6,1,1,1,2,1,1,1,2,3,1,1,4,7,5,1,2,1,3,1,2,1,1,1,2,
%U A000188 1,1,3,8,1,1,1,2,1,1,1,6,1,1,5,2,1,1,1,4,9,1,1,2,1,1,1,2,1,3
%N A000188 (1) Number of solutions to x^2 = 0 (mod n). (2) Also square root of largest square dividing n. (3) Also Max_{ d divides n } GCD[d,n/d].
%C A000188 Shadow transform of the squares A000290. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 02 2002
%C A000188 Labos Elemer (LABOS(AT)ana.sote.hu) and Henry Bottomley (se16(AT)btinternet.com) independently proved that (2) and (3) define the same sequence. Bottomley also showed that (1) and (2) define the same sequence.
%C A000188 Labos: Proof that (2)=(3): Let Max{[GCD[d,n/d]}=K, then d=Kx,n/d=Ky so n=KKxy where xy is the square-free part of n,otherwise K is not maximal. Observe also that g=GCD[K,xy] is not necessarily 1. Thus K is also the "maximal square-root factor" of n.
%C A000188 A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), LCM(b,c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n.
%D A000188 Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, 2008, to appear.
%H A000188 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000188.txt">Table of n, a(n) for n=1..10000</a>
%H A000188 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">Some Smarandache-type multiplicative sequences</a>
%H A000188 Kevin A. Broughan, <a href="http://www.math.waikato.ac.nz/~kab/div4.pdf">Restricted divisor sums</a>, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
%H A000188 S. R. Finch and Pascal Sebah, <a href="http://arXiv.org/abs/math.NT/0604465">Squares and Cubes Modulo n</a> (arXiv:math.NT/0604465).
%H A000188 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%F A000188 a(n) = Sum_{d^2|n} phi(d), where phi = Euler totient function A000010.
%F A000188 Multiplicative with a(p^e) = p^[e/2]. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A000188 Dirichlet series: Sum(n=1..inf, a(n)/n^s) = zeta(2*s-1)*zeta(s)/zeta(2*s), (Re(s)>1).
%p A000188 with(numtheory):A000188 := proc(n) local i: RETURN(op(mul(i,i=map(x->x[1]^floor(x[2]/2),ifactors(n)[2])))); end;
%t A000188 Array[ Function[ n, Count[ Array[ PowerMod[ #, 2, n ]&, n, 0 ], 0 ] ], 100 ]
%o A000188 (PARI) a(n)=if(n<1,0,sum(i=1,n,i*i%n==0))
%Y A000188 a(n) = n/A019554(n)
%Y A000188 Cf. A008833, A007913, A117811, A046951, A055210.
%Y A000188 Cf. A007913, A007947, A019554. For partial sums see A120486.
%Y A000188 Sequence in context: A046854 A066170 A071773 this_sequence A097886 A088863 A053283
%Y A000188 Adjacent sequences: A000185 A000186 A000187 this_sequence A000189 A000190 A000191
%K A000188 nonn,easy,nice,mult
%O A000188 1,4
%A A000188 njas

%I A097886
%S A097886 1,1,1,1,2,1,1,1,2,3,1,1,2,3,5,1,2,3,5,7,2,3,5,7,10,1,1,1,2,3,1,1,2,3,5,
%T A097886 1,2,3,5,7,2,3,5,7,10,3,5,7,10,14,1,1,2,3,5,1,2,3,5,7,2,3,5,7,10,3,5,7,
%U A097886 10,14,5,7,10,14,20,1,2,3,5,7,2,3,5,7,10,3,5,7,10,14,5,7,10,14,20,7,10
%N A097886 Near-minimal Pisot (near theta2) 5 X 5 Markov sequence.
%F A097886 M={{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 0, -1, 1, 1}} A[n_]:=M.A[n-1]; A[0]:={{1, 1, 1, 1, 1}, {1, 1, 1, 1, 2}, {1, 1, 1, 2, 3}, {1, 1, 2, 3, 5}, {1, 2, 3, 5, 7}};
%t A097886 Clear[M, A, x] (* near Minimal Pisot ( near theta2) 5 X 5 Markov sequence*) digits=12 M={{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 0, -1, 1, 1}} Det[M] A[n_]:=M.A[n-1]; A[0]:={{1, 1, 1, 1, 1}, {1, 1, 1, 1, 2}, {1, 1, 1, 2, 3}, {1, 1, 2, 3, 5}, {1, 2, 3, 5, 7}}; i=IdentityMatrix[5] Det[M-x*i] (* flattened sequence of 5 X 5 matrices made with a (near theta2) Minimal Pisot recurrence*) b=Flatten[Table[M.A[n], {n, 0, digits}]] Dimensions[b][[1]] ListPlot[b, PlotJoined->True]
%Y A097886 Sequence in context: A066170 A071773 A000188 this_sequence A088863 A053283 A035669
%Y A097886 Adjacent sequences: A097883 A097884 A097885 this_sequence A097887 A097888 A097889
%K A097886 nonn,uned
%O A097886 0,5
%A A097886 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 02 2004

%I A088863
%S A088863 1,1,1,1,2,1,1,1,2,3,1,2,2,3,3,3,2,1,2,3,3,3,2,1,2,2,2,1,2,5,1,2,2,2,2,
%T A088863 5,4,5,2,4,3,4,5,3,2,2,3,6,2,4,4,6,2,5,3,4,2,2,3,2,3,2,5,3,4,4,3,5,2,3,
%U A088863 3,6,5,2,2,5,3,9,4,3,5,2,8,4,4,3,5,2,4,6,3,4,2,7,3,4,4,1,2,5,4,5,3,5,4
%N A088863 Number of prime factors of n-th Mersenne number M(p_n).
%C A088863 a(n) = A001222(A001348(n)) = A001222(A000225(A000040(n))
%H A088863 Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007, <a href="http://www.research.att.com/~njas/sequences/b088863.txt">Table of n, a(n) for n = 1..137</a>
%H A088863 Herman Jamke, <a href="http://www.research.att.com/~njas/sequences/a088863.txt">The first 137 terms in detail</a>
%e A088863 a(5)=2 because M(p_5)=M(11)=2047 has 2 (not necessarily distinct) prime factors.
%p A088863 seq(nops(ifactor(2^ithprime(n)-1)),n=1..32); (Deutsch)
%t A088863 Do[m = 2^Prime[n] - 1; Print[Plus @@ Last /@ FactorInteger[m]], {n, 1, 50}] (Propper)
%o A088863 (PARI) for(n=1,137,print1(bigomega(2^prime(n)-1)",")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007
%Y A088863 Cf. A046051, A001348.
%Y A088863 Sequence in context: A071773 A000188 A097886 this_sequence A053283 A035669 A126863
%Y A088863 Adjacent sequences: A088860 A088861 A088862 this_sequence A088864 A088865 A088866
%K A088863 nonn
%O A088863 1,5
%A A088863 Jeppe Stig Nielsen (mail(AT)jeppesn.dk), Nov 25 2003
%E A088863 14 more terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 23 2004
%E A088863 More terms from Ryan Propper (rpropper(AT)stanford.edu), Jul 31 2005
%E A088863 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

%I A053283
%S A053283 1,1,1,0,1,2,1,1,1,2,3,1,2,4,3,2,3,5,4,4,5,6,7,5,6,9,9,7,9,12,11,11,
%T A053283 12,15,16,14,16,21,20,18,22,25,26,25,28,33,34,33,35,42,43,41,47,53,53,
%U A053283 54,57,65,69,67,73,83,85,83,92,102,104,106,114,125,130,130,139,154
%V A053283 1,-1,1,0,1,-2,1,-1,1,-2,3,-1,2,-4,3,-2,3,-5,4,-4,5,-6,7,-5,6,-9,9,-7,9,-12,11,-11,
%W A053283 12,-15,16,-14,16,-21,20,-18,22,-25,26,-25,28,-33,34,-33,35,-42,43,-41,47,-53,53,
%X A053283 -54,57,-65,69,-67,73,-83,85,-83,92,-102,104,-106,114,-125,130,-130,139,-154
%N A053283 Coefficients of the '10th order' mock theta function X(q)
%D A053283 Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) 497-569
%D A053283 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9
%F A053283 G.f.: X(q) = sum for n >= 0 of (-1)^n q^n^2/((1+q)(1+q^2)...(1+q^(2n)))
%t A053283 Series[Sum[(-1)^n q^n^2/Product[1+q^k, {k, 1, 2n}], {n, 0, 10}], {q, 0, 100}]
%Y A053283 Other '10th order' mock theta functions are at A053281, A053282, A053284.
%Y A053283 Sequence in context: A000188 A097886 A088863 this_sequence A035669 A126863 A106806
%Y A053283 Adjacent sequences: A053280 A053281 A053282 this_sequence A053284 A053285 A053286
%K A053283 sign,easy
%O A053283 0,6
%A A053283 Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 19 1999

%I A035669
%S A035669 0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,2,1,1,1,2,3,1,4,3,3,3,5,7,4,7,7,8,8,10,
%T A035669 13,10,14,15,16,17,20,25,21,26,29,32,33,37,45,41,47,54,58,61,65,79,76,
%U A035669 83,94,103,108,113,132,135,143,160,172,185,192,219,227,240,265,286
%N A035669 Number of partitions of n into parts 7k+4 and 7k+5 with at least one part of each type.
%Y A035669 Sequence in context: A097886 A088863 A053283 this_sequence A126863 A106806 A039958
%Y A035669 Adjacent sequences: A035666 A035667 A035668 this_sequence A035670 A035671 A035672
%K A035669 nonn
%O A035669 1,16
%A A035669 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A126863
%S A126863 1,1,2,1,1,1,2,3,2,1,1,1,2,1,1,1,2,3,4,3,2,1,1,1,2,1,1,1,2,3,2,1,1,1,2,
%T A126863 1,1,1,2,3,4,5,4,3,2,1,1,1,2,1,1,1,2,3,2,1,1,1,2,1,1,1,2,3,4,3,2,1,1,1,
%U A126863 2,1,1,1,2,3,2,1,1,1,2,1,1,1,2,3,4,5,6,5,4,3,2,1,1,1,2,1,1,1,2,3,2,1,1
%N A126863 S(1)={1}. S(n) = {S(n-1) {1,2,...,(n-1),n,(n-1),...,2,1} S(n-1)}, where S(n) is a string of the first 3*2^n -2*n -3 terms of the sequence.
%C A126863 S(n) consists of the first 3*2^n - 2*n - 3 terms of the sequence, i.e. S(n) consists of A050488(n) terms. Each S(n) forms a palindrome.
%e A126863 S(1) = {1}. S(2) = {1,1,2,1,1}. S(3) = {1,1,2,1,1,1,2,3,2,1,1,1,2,1,1}. S(4) = {1,1,2,1,1,1,2,3,2,1,1,1,2,1,1,1,2,3,4,3,2,1,1,1,2,1,1,1,2,3,2,1,1,1,2,1,1}, etc.
%Y A126863 Cf. A050488.
%Y A126863 Sequence in context: A088863 A053283 A035669 this_sequence A106806 A039958 A029344
%Y A126863 Adjacent sequences: A126860 A126861 A126862 this_sequence A126864 A126865 A126866
%K A126863 easy,nonn
%O A126863 1,3
%A A126863 Leroy Quet (qq-quet(AT)mindspring.com), Mar 15 2007

%I A106806
%S A106806 1,1,1,2,1,1,1,2,3,2,1,1,1,2,3,3,3,1,4,2,4,1,1,5,6,2,4,3,1,7,3,1,1,
%T A106806 2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,10,9,2
%N A106806 If the name of the n-th King or Queen of Denmark is King (or Queen) XXX the k-th, then the n-th term is k.
%H A106806 Robert Warholm, <a href="http://www.warholm.nu/Kingdan.html">All the Kings of Denmark</a>
%e A106806 According to Warholm's website, the current monarch, Queen Margrethe II, is the 52-nd ruler of Denmark, so a(52) = 2.
%Y A106806 Cf. A113515.
%Y A106806 Sequence in context: A053283 A035669 A126863 this_sequence A039958 A029344 A125769
%Y A106806 Adjacent sequences: A106803 A106804 A106805 this_sequence A106807 A106808 A106809
%K A106806 nonn,fini
%O A106806 1,4
%A A106806 njas, Feb 21 2006

%I A039958
%S A039958 1,1,1,2,1,1,1,2,3,2,1,2,1,1,2,4,1,3,1,2,1,1,1,2,5,2,3,2,1,2,1,4,1,
%T A039958 2,2,6,1,1,2,2,1,2,1,2,3,1,1,4,7,5,2,2,1,3,2,2,1,2,1,2,1,1,3,8,2,2,
%U A039958 1,2,1,2,1,6,1,2,5,2,1,2,3,4,9,4,1,2,2,1,2,2,1,3,2,2,1,1,2,4,1,7,3,10
%V A039958 1,-1,1,2,-1,1,1,2,3,-2,1,2,-1,1,2,4,-1,3,1,2,1,1,1,2,5,-2,3,2,-1,2,1,4,1,
%W A039958 2,2,6,-1,1,2,2,-1,2,1,2,3,1,1,4,7,5,2,2,-1,3,2,2,1,-2,1,2,-1,1,3,8,-2,2,
%X A039958 1,2,1,2,1,6,-1,-2,5,2,1,2,3,4,9,-4,1,2,-2,1,2,2,-1,3,2,2,1,1,2,4,-1,7,3,10
%N A039958 Class number of maximal order in real quadratic field of radicand n.
%D A039958 R. A. Mollin, Quadratics, CRC Press, 1996, Table C1.
%Y A039958 Sequence in context: A035669 A126863 A106806 this_sequence A029344 A125769 A003023
%Y A039958 Adjacent sequences: A039955 A039956 A039957 this_sequence A039959 A039960 A039961
%K A039958 sign,easy,nice
%O A039958 1,4
%A A039958 R. K. Guy (rkg(AT)cpsc.ucalgary.ca), njas
%E A039958 Mollin gives a table for n < 10000.

%I A029344
%S A029344 1,0,0,0,1,1,0,0,1,1,2,1,1,1,2,3,2,1,2,3,5,3,3,3,5,6,5,
%T A029344 4,5,6,9,7,7,7,9,11,10,9,10,11,15,13,13,13,16,18,17,16,
%U A029344 18,19,23,21,22,22,25,28,27,26,28,30,35,32,33,34,38,41
%N A029344 Expansion of 1/((1-x^4)(1-x^5)(1-x^10)(1-x^11)).
%Y A029344 Sequence in context: A126863 A106806 A039958 this_sequence A125769 A003023 A114731
%Y A029344 Adjacent sequences: A029341 A029342 A029343 this_sequence A029345 A029346 A029347
%K A029344 nonn
%O A029344 0,11
%A A029344 njas

%I A125769
%S A125769 1,2,1,1,1,2,3,2,4,1,2,1,2,2,8,9,4,4,1,2,2,1,3,2,16,10,17,3,2,4,6,2,1,5,
%T A125769 10,10,2,27,6,29,4,2,1,32,3,3,1,8,38,23,3,2,2,7,43,4,6,2,1,10,47,14,2,4,
%U A125769 4,53,5,12,58,35,59,3,61,62,1,64,5,6,40,3,2,2,12,12,8,74,10,76,2,2,6,4
%N A125769 a(n) is the least number j such that j*T_k +/- 1 is n-th prime for some k-th triangular number.
%C A125769 Eventually all primes p appear since (p +/-1) times 1(1+1)/2 equals (p +/- 1).
%C A125769 If we asked for the least number k then k always equals 1 since all primes p appear since (p +/-1) times 1(1+1)/2 equals (p +/- 1).
%C A125769 The k's for the corresponding j's are: round(sqrt(2p/j)).
%C A125769 First occurrence of i is A125770: 1, 2, 7, 9, 34, 31, 54, 15, 16, 26, 148, 68, 398, 62, 193, 25, 27, 140, 550, 397, 107, 113, ...,.
%e A125769 a(1) = 1 because 1*1+1 = 2 which is the first prime,
%e A125769 a(2) = 2 because 2*1+1 = 3 which is the second prime,
%e A125769 a(3) = 4 because 1*6-1 = 5 which is the third prime,
%e A125769 a(8) = 3 because 2*10-1 = 19 which is the eighth prime, ...
%t A125769 triQ[n_] := IntegerQ@ Sqrt[8n + 1]; f[n_] := Block[{j = 1, p = Prime@n}, While[ !triQ[(p - 1)/j] && !triQ[(p + 1)/j], j++ ]; j]; Array[f, 92]
%Y A125769 Cf. A000217, A125765, A125766, A125767, A125768, A125770.
%Y A125769 Sequence in context: A106806 A039958 A029344 this_sequence A003023 A114731 A035389
%Y A125769 Adjacent sequences: A125766 A125767 A125768 this_sequence A125770 A125771 A125772
%K A125769 nonn
%O A125769 1,2
%A A125769 Jonathan Vos Post (jvospost2(AT)yahoo.com) & Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 01 2006

%I A003023 M0062
%S A003023 0,1,1,2,1,1,1,2,3,3,1,6,1,4,4,5,1,3,1,6,2,5,1,4,2,6,2,1,1,14,1,2,5,7,2,
%T A003023 3,1,6,2,3,1,13,1,4,6,7,1,5,3,2,3,8,1,12,2,4,2,3,1,10,1,8,2,3,2,11,1,4,
%U A003023 3,5,1,8,1,4,4,4,2,10,1,6,4,5,1,5,2,8,6,6,1,9,3,5,3,3,3,8,1,2,3,4,1,17
%N A003023 "Length" of aliquot sequence for n.
%C A003023 The aliquot sequence for n is the trajectory of n under repeated application of the map x -> sigma(x) - x.
%C A003023 The trajectory will either have a transient part followed by a cyclic part, or will have an infinite transient part and never cycle.
%C A003023 Sequence gives (length of transient part of trajectory) - 1 + (length of cycle provided cycle is nonzero). See A098007 for a better version.
%C A003023 Examples of trajectories:
%C A003023 1, 0, 0, ...
%C A003023 2, 1, 0, 0, ...
%C A003023 3, 1, 0, 0, ... (and similarly for any prime)
%C A003023 4, 3, 1, 0, 0, ...
%C A003023 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%C A003023 6, 6, 6, ... (and similarly for any perfect number)
%C A003023 8, 7, 1, 0, 0, ...
%C A003023 9, 4, 3, 1, 0, 0, ...
%C A003023 12, 16, 15, 9, 4, 3, 1, 0, 0, ...
%C A003023 14, 10, 8, 7, 1, 0, 0, ...
%C A003023 25, 6, 6, 6, ...
%C A003023 28, 28, 28, ... (the next perfect number)
%C A003023 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...
%C A003023 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...
%D A003023 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A003023 R. K. Guy, Unsolved Problems in Number Theory, B6.
%D A003023 R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.
%H A003023 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AliquotSequence.html">Link to a section of The World of Mathematics.</a>
%H A003023 Matthew M. Conroy, <a href="http://www.madandmoonly.com/doctormatt">Home page</a> (listed instead of email address)
%H A003023 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/aliquot.htm">Aliquot series:Abundant,deficient,perfect</a>
%p A003023 f:=proc(n) local t1, i,j,k; t1:=[n]; for i from 2 to 50 do j:= t1[i-1]; k:=sigma(j)-j; t1:=[op(t1), k]; od: t1; end; # produces trajectory for n
%o A003023 (MuPAD) s := func(_plus(op(numlib::divisors(n)))-n,n): A003023 := proc(n) local i,T,m; begin m := n; i := 1; while T[ m ]<>1 and m<>1 do T[ m ] := 1; m := s(m); i := i+1 end_while; i-1 end_proc:
%Y A003023 Cf. A098007.
%Y A003023 Cf. A059447 (least k such that n is the length of the aliquot sequence for k).
%Y A003023 Sequence in context: A039958 A029344 A125769 this_sequence A114731 A035389 A129176
%Y A003023 Adjacent sequences: A003020 A003021 A003022 this_sequence A003024 A003025 A003026
%K A003023 nonn,easy
%O A003023 1,4
%A A003023 njas
%E A003023 More terms from Matthew Conroy (list1(AT)madandmoonly.com), Jan 16 2006

%I A114731
%S A114731 1,2,1,1,1,2,3,3,2,2,1,1,1,2,3,4,5,4,4,3,3,2,2,1,1,1,2,3,4,5,6,6,5,5,4,
%T A114731 4,3,3,2,2,1,1,1,2,3,4,5,6,7,8,7,7,6,6,5,5,4,4,3,3,2,2,1,1,1,2,3,4,5,6,
%U A114731 7,8,9,9,8,8,7,7,6,6,5,5,4,4,3,3,2,2,1,1,1,2,3,4,5,6,7,8,9,10,11,10,10
%N A114731 One of a family of six fractal sequences that transform into each other.
%C A114731 Let A be the sequence A114729 (1, 1, 1, 2, 3, 2, 2, 1, 1, 1, ...), B be the sequence A114730 (1, 1, 2, 2, 1, 1, 1, 2, 3, 4, ...) and C be the sequence A114731 (1, 2, 1, 1, 1, 2, 3, 3, 2, 2, ...). Let D be the sequence A114732 (1, 2, 3, 1, 1, 2, 3, 4, 5, 6, ...), E be the sequence A114733 (1, 2, 1, 2, 3, 4, 5, 3, 1, 1, ...) and F be the sequence A114734 (1, 1, 2, 3, 4, 2, 1, 2, 3, 4, ...). Then:
%C A114731 - A upper trims to B
%C A114731 - B upper trims to C
%C A114731 - C upper trims to A
%C A114731 - A lower trims to C
%C A114731 - B lower trims to A
%C A114731 - C lower trims to B
%C A114731 - D gives the number of times each element of A occurs
%C A114731 - E gives the number of times each element of B occurs
%C A114731 - F gives the number of times each element of C occurs
%C A114731 - A gives the number of times each element of D occurs
%C A114731 - B gives the number of times each element of E occurs
%C A114731 - C gives the number of times each element of F occurs
%C A114731 - D lower trims to E
%C A114731 - E lower trims to F
%C A114731 - F lower trims to D
%C A114731 - D upper trims to F
%C A114731 - E upper trims to D
%C A114731 - F upper trims to E
%e A114731 F(10)=4 and that's the second 4 in that sequence, so C(10)=2.
%Y A114731 Cf. A114729, A114730, A114732, A114733, A114734.
%Y A114731 Sequence in context: A029344 A125769 A003023 this_sequence A035389 A129176 A134132
%Y A114731 Adjacent sequences: A114728 A114729 A114730 this_sequence A114732 A114733 A114734
%K A114731 nonn
%O A114731 1,2
%A A114731 Kerry Mitchell (lkmitch(AT)gmail.com), Dec 28 2005

%I A035389
%S A035389 0,0,0,1,1,0,0,1,1,2,1,1,1,2,3,3,2,2,3,6,5,5,4,6,8,9,9,9,9,13,14,17,16,
%T A035389 17,19,23,26,28,28,32,34,41,44,49,50,54,60,70,75,81,83,93,102,116,124,
%U A035389 132,138,153,169,189,198,211,224,250,273,298,313,336,359,397,429
%N A035389 Number of partitions of n into parts 6k+4 or 6k+5.
%Y A035389 Sequence in context: A125769 A003023 A114731 this_sequence A129176 A134132 A030424
%Y A035389 Adjacent sequences: A035386 A035387 A035388 this_sequence A035390 A035391 A035392
%K A035389 nonn
%O A035389 1,10
%A A035389 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A129176
%S A129176 1,1,1,1,1,2,1,1,1,2,3,3,3,1,1,1,2,3,5,5,7,7,6,4,1,1,1,2,3,5,7,9,11,14,
%T A129176 16,16,17,14,10,5,1,1,1,2,3,5,7,11,13,18,22,28,32,37,40,44,43,40,35,25,
%U A129176 15,6,1,1,1,2,3,5,7,11,15,20,26,34,42,53,63,73,85,96,106,113,118,118
%N A129176 Triangle read by rows: T(n,k) is the number of Dyck words of length 2n having k inversions (n>=0, k>=0). A Catalan word of length 2n is a word of n 0's and n 1's for which no initial segment contains more 1's than 0's.
%C A129176 Representing a Dyck word p of length 2n as a superdiagonal Dyck path p', the number of inversions of p is equal to the area between p' and the path that corresponds to the Dyck word 0^n 1^n. Row n has 1+n(n-1)/2 terms. Row sums are the Catalan numbers (A000108). Alternating row sums for n>=1 are the Catalan numbers alternated with 0's (A097331). Sum(k*T(n,k),k>=0)=A029760(n-2). A modified form of A129182 (area under Dyck paths).
%C A129176 Comment from Alford Arnold, Jan 29 2008: This triangle gives the partial sums of the following triangle:
%C A129176 1
%C A129176 .1
%C A129176 ....2...1
%C A129176 ........2...3...3...1
%C A129176 ............2...2...6...7...6...4...1
%C A129176 ................2...2...4...8..12..15..17..14..10...5...1
%C A129176 etc.
%D A129176 J. Furlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, A, 40, 248-264, 1985.
%D A129176 M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.
%F A129176 The row generating polynomials P[n]=P[n](t) satisfy P[0]=1, P[n+1]=Sum(t^((i+1)(n-i))P[i]P[n-i],i=0..n).
%e A129176 T(4,5)=3 because we have 01010011, 01001101 and 00110101.
%e A129176 Triangle starts:
%e A129176 1;
%e A129176 1;
%e A129176 1,1;
%e A129176 1,1,2,1;
%e A129176 1,1,2,3,3,3,1;
%e A129176 1,1,2,3,5,5,7,7,6,4,1;
%p A129176 P[0]:=1: for n from 0 to 8 do P[n+1]:=sort(expand(sum(t^((i+1)*(n-i))*P[i]*P[n-i],i=0..n))) od: for n from 1 to 9 do seq(coeff(P[n],t,j),j=0..n*(n-1)/2) od; # yields sequence in triangular form
%Y A129176 Cf. A000108, A097331, A029760, A129182.
%Y A129176 Cf. A136624 A136625.
%Y A129176 Sequence in context: A003023 A114731 A035389 this_sequence A134132 A030424 A026519
%Y A129176 Adjacent sequences: A129173 A129174 A129175 this_sequence A129177 A129178 A129179
%K A129176 nonn,tabf
%O A129176 0,6
%A A129176 Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 11 2007

%I A134132
%S A134132 1,1,1,0,0,1,1,2,1,1,1,2,3,3,3,2,2,3,5,6,5,4,4,6,9,10,9,8,8,11,14,16,15,
%T A134132 13,14,18,24,26,25,22,23,29,36,40,38,36,38,46,56,61,60,56,59,70,84,92,
%U A134132 90,86,90,106,125,135,134,130,136,157,181,196,195,191,201,228,263,282
%N A134132 Expansion of q^(-1/6) * eta(q^2) * eta(q^3)^2 * eta(q^18) / ( eta(q) * eta(q^6)^2 * eta(q^9) ) in powers of q.
%F A134132 Expansion of (chi(-q) * chi(-q^9) / chi(-q^3)^2)^(-1) in power of q where chi() is a Ramanujan theta function.
%F A134132 Euler transform of period 18 sequence [ 1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, 1, 0, ...].
%F A134132 Given g.f. A(x) then B(x) = A(x^6) * x satisfies 0 = f(B(x), B(x^2), B(x^4) ) where f(u, v, w) = (u^2 + v) * w^2 - (u^2 - v) * v.
%F A134132 Given g.f. A(x) then B(x) = A(x^3)^2 * x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) * (1 + u * v) - (2 * u * v)^2.
%F A134132 G.f. is a period 1 Fourier series which satisfies f(-1 / (648 t)) = 1 / f(t) where q = exp(2 pi i t).
%F A134132 G.f.: Product_{k>0} (1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2.
%e A134132 q + q^7 + q^13 + q^31 + q^37 + 2*q^43 + q^49 + q^55 + q^61 + 2*q^67 + ...
%o A134132 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^18 + A) / ( eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A) ), n))}
%Y A134132 A112178(3*n+1) = -a(n). Convolution inverse of A134131.
%Y A134132 Sequence in context: A114731 A035389 A129176 this_sequence A030424 A026519 A025177
%Y A134132 Adjacent sequences: A134129 A134130 A134131 this_sequence A134133 A134134 A134135
%K A134132 nonn
%O A134132 0,8
%A A134132 Michael Somos, Oct 10 2007, Oct 21 2007

%I A030424
%S A030424 1,0,0,1,2,1,1,1,2,3,4,5,4,4,3,3,4,4,5,6,5,6,5,6,6,6,6,5,4,5,
%T A030424 6,7,7,6,7,6,7,6,6,5,5,5,6,5,5,6,5,5,5,5,6,5,4,4,5,4,4,5,5,5,
%U A030424 6,7,8,9,10,9,8,9,8,8,9,8,7,6,6,6,6,5,6,6,7,8,8,7,7,7,8,7,7,7
%N A030424 (# 1's)-(# 2's) in first n terms of A030413.
%Y A030424 Sequence in context: A035389 A129176 A134132 this_sequence A026519 A025177 A026148
%Y A030424 Adjacent sequences: A030421 A030422 A030423 this_sequence A030425 A030426 A030427
%K A030424 nonn
%O A030424 1,5
%A A030424 Clark Kimberling (ck6(AT)evansville.edu)

%I A026519
%S A026519 1,1,1,1,1,1,2,1,1,1,2,4,4,4,2,1,1,2,5,6,8,6,5,2,1,1,3,8,13,19,
%T A026519 20,19,13,8,3,1,1,3,9,16,27,33,38,33,27,16,9,3,1,1,4,13,28,52,
%U A026519 76,98,104,98,76,52,28,13,4,1,1,4,14,32,65,104,150,180,196,180
%N A026519 Triangular array T read by rows: T(i,0)=T(i,2i)=1 for i >= 0, T(i,1)=T(i,2i-1)=[ (i+1)/2 ] for i >= 1; for even n >= 2, T(i,j)=T(i-1,j-2)+T(i-1,j) for 2<=j<=2i-2; for odd n >= 3, T(i,j)=T(i-1,j-2)+T(i-1,j-1)+T(i-1,j) for 2<=j<=2i-2.
%F A026519 T(n, k) = number of integer strings s(0), ..., s(n) such that s(0)=0, s(n)=n-k, |s(i)-s(i-1)|=1 if i is even, |s(i)-s(i-1)|<= 1 if i is odd.
%Y A026519 Cf. A026527.
%Y A026519 Sequence in context: A129176 A134132 A030424 this_sequence A025177 A026148 A117211
%Y A026519 Adjacent sequences: A026516 A026517 A026518 this_sequence A026520 A026521 A026522
%K A026519 nonn,tabl
%O A026519 1,7
%A A026519 Clark Kimberling (ck6(AT)evansville.edu)

%I A025177
%S A025177 1,1,0,1,1,1,2,1,1,1,2,4,4,4,2,1,1,3,7,10,12,10,7,3,1,1,4,11,20,29,32,
%T A025177 29,20,11,4,1,1,5,16,35,60,81,90,81,60,35,16,5,1,1,6,22,56,111,176,231,
%U A025177 252,231,176,111,56,22,6,1,1,7,29,84,189,343,518,659,714,659,518,343
%N A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.
%F A025177 T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].
%F A025177 G.f.: (1-yz)/[1-z(1+y+y^2)].
%e A025177 .............1
%e A025177 ..........1..0..1
%e A025177 .......1..1..2..1..1
%e A025177 ....1..2..4..4..4..2..1
%e A025177 ..1.3..7..10.12.10.7..3..1
%e A025177 1.4.11.20.29.32.29.20.11.4.1
%o A025177 (PARI) T(n,k)=if(n<0||k<0||k>2*n,0,if(n==0,1,if(n==1,[1,0,1][k+1],if(n==2,[1,1,2,1,1][k+1],T(n-1,k-2)+T(n-1,k-1)+T(n-1,k)))))
%o A025177 (PARI) T(n,k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)
%o A025177 (PARI) {T(n, k)= if(n<0||k<0||k>2*n, 0, if(n==0, 1, polcoeff( (1+x+x^2)^n, k)- polcoeff( (1+x+x^2)^(n-1), k-1)))}
%Y A025177 Columns include A025178, A025179, A025180, A025181, A025182.
%Y A025177 Cf. A024996.
%Y A025177 Sequence in context: A134132 A030424 A026519 this_sequence A026148 A117211 A061545
%Y A025177 Adjacent sequences: A025174 A025175 A025176 this_sequence A025178 A025179 A025180
%K A025177 nonn,tabf,easy
%O A025177 1,7
%A A025177 Clark Kimberling (ck6(AT)evansville.edu)
%E A025177 Edited by Ralf Stephan, Jan 09 2005

%I A026148
%S A026148 1,1,1,1,1,2,1,1,1,2,4,4,4,2,1,3,7,10,12,10,6,1,4,11,20,29,32,28,16,1,5,16,
%T A026148 35,60,81,89,76,44,1,6,22,56,111,176,230,246,209,120,1,7,29,84,189,343,517,
%U A026148 652,685,575,329,1,8,37,120,302,616,1049,1512,1854,1912,1589,904,1,9,46,165
%N A026148 Triangular array T read by rows: T(i,0)=1 for i >= 0, T(1,1)=1,T(2,1)=1,T(2,2)=2,T(2,3)=1,T(2,4)= 1, and for i >= 3, T(i,1)=i-1, T(i,i+2)=T(i-1,i)+T(i-1,i-1), T(i,j)=T(i-1,j-2)+T(i-1,j-1)+T(i-1,j) for j=2,3,....,i+1. For n >= 2, T(n,k)=number of nonneg. int. strings s(0),...,s(n) such that s(n)=n-k, s(0)=2, |s(1)-2|=1, and {s(i)-s(i-1)|<=1 for i >= 2.
%Y A026148 Sequence in context: A030424 A026519 A025177 this_sequence A117211 A061545 A126886
%Y A026148 Adjacent sequences: A026145 A026146 A026147 this_sequence A026149 A026150 A026151
%K A026148 nonn,tabl
%O A026148 1,6
%A A026148 Clark Kimberling (ck6(AT)evansville.edu)

%I A117211
%S A117211 1,1,2,1,1,1,2,4,4,4,3,2,0,1,2,3,4,5,5,4,4,3,1,1,2,3,5,5,5,3,1,1,3,4,3,
%T A117211 2,2,1,3,4,6,4,4,5,0,4,2,1,4,2,3,3,6,9,7,1,1,4,8,10,6,10,11,12,9,4,7,7,
%U A117211 15,25,10,5,13,1,6,16,21,14,15,28,6,12,3,1,18,18,17,25,13
%V A117211 1,-1,2,-1,1,1,-2,4,-4,4,-3,2,0,-1,2,-3,4,-5,5,-4,4,-3,1,1,-2,3,-5,5,-5,3,-1,1,3,-4,3,
%W A117211 -2,2,-1,-3,4,-6,4,-4,5,0,-4,2,-1,4,-2,3,-3,6,-9,7,-1,1,-4,-8,10,-6,10,-11,12,-9,-4,7,
%X A117211 -7,15,-25,10,-5,13,1,-6,16,-21,14,-15,28,-6,-12,-3,1,18,-18,17,-25,13
%N A117211 G.f. A(x) satisfies: 1/(1+x) = product_{n>=1} A(x^n).
%C A117211 Self-convolution inverse is A117210.
%F A117211 G.f.: A(x) = exp( -Sum_{n>=1} A117212(n)*x^n/n ).
%F A117211 G.f.: A(x) = product_{k>=1}(1 + x^k)^(-mu(k)) where mu(k) is the Moebius (Mobius) function, A008683 - Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006
%t A117211 nmax = 88; CoefficientList[ Series[ Product[ (1 + x^k)^(-MoebiusMu[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ] - Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006
%o A117211 (PARI) {a(n)=if(n==0,1,if(n==1,-1, (-1)^n-polcoeff(prod(i=1,n,sum(k=0,min(n\i,n-1),a(k)*x^(i*k))+x*O(x^n)),n,x)))}
%Y A117211 Cf. A117212 (log.g.f.), A117210 (inverse); variants: A117208, A117209.
%Y A117211 Sequence in context: A026519 A025177 A026148 this_sequence A061545 A126886 A105685
%Y A117211 Adjacent sequences: A117208 A117209 A117210 this_sequence A117212 A117213 A117214
%K A117211 sign
%O A117211 0,3
%A A117211 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 03 2006

%I A061545
%S A061545 0,2,1,1,1,2,4,16
%N A061545 Minimal XR-sequences of A048200(n).
%C A061545 a(5) = 1 since "xrrxrrxrxr" is the shortest only sequence reversing "ABCDE".
%e A061545 For n = 6 there are 2 possibilities: A048200(6) = 15. -> Two minimal XR-sequences with length 15 exist. -> "xrxrrxrxrrxrrrx" "rxrrxrxrrxrxrrx"; so a(6) = 2.
%Y A061545 Cf. A048200.
%Y A061545 Sequence in context: A025177 A026148 A117211 this_sequence A126886 A105685 A110858
%Y A061545 Adjacent sequences: A061542 A061543 A061544 this_sequence A061546 A061547 A061548
%K A061545 more,nonn
%O A061545 1,2
%A A061545 Naohiro Nomoto (6284968128(AT)geocities.co.jp), May 16 2001

%I A126886
%S A126886 1,1,1,1,1,1,1,2,1,1,1,2,5,6,8,7,7,5,5,3,2,1,1,1,2,7,12,23,30,42,47,
%T A126886 55,53,53,45,40,29,23
%N A126886 Triangle read by rows, arising in enumeration of 4-trees by diameter.
%C A126886 See reference for precise definition.
%D A126886 A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (No. 4, 1988), 304-313.
%e A126886 Triangle begins:
%e A126886 1
%e A126886 1
%e A126886 1 1 1
%e A126886 1 1 2 1 1
%e A126886 1 2 5 6 8 7 7 5 5 3 2 1 1
%e A126886 1 2 7 12 23 30 42 47 55 53 53 45 40 29 23 ...
%Y A126886 Cf. A000602 (diagonal sums when rows are displaced by the appropriate amounts), A036437.
%Y A126886 Sequence in context: A026148 A117211 A061545 this_sequence A105685 A110858 A125090
%Y A126886 Adjacent sequences: A126883 A126884 A126885 this_sequence A126887 A126888 A126889
%K A126886 nonn,tabf
%O A126886 0,8
%A A126886 njas, Aug 04 2007

%I A125090
%S A125090 1,1,0,1,1,1,1,2,1,1,1,3,0,3,0,1,4,2,5,2,1,1,5,5,6,7,2,1,1,6,9,5,15,0,5,0,
%T A125090 1,7,14,1,25,9,12,3,1,1,8,20,7,35,29,18,15,3,1,1,9,27,20,42,63,14,42,0,7,
%U A125090 0,1,10,35,39,42,112,14,85,24,22,4,1,1,11,44,65,30,174,84,134,95,40,26,4
%V A125090 1,1,0,1,-1,-1,1,-2,-1,1,1,-3,0,3,0,1,-4,2,5,-2,-1,1,-5,5,6,-7,-2,1,1,-6,9,5,-15,0,5,0,
%W A125090 1,-7,14,1,-25,9,12,-3,-1,1,-8,20,-7,-35,29,18,-15,-3,1,1,-9,27,-20,-42,63,14,-42,0,7,
%X A125090 0,1,-10,35,-39,-42,112,-14,-85,24,22,-4,-1,1,-11,44,-65,-30,174,-84,-134,95,40,-26,-4
%N A125090 Triangle read by rows: T(0,0)=1; for 0<=k<=n, n>=1, T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the tridiagonal n X n matrix with diagonal (0,1,1,...) and super- and subdiagonals (1,1,1,...).
%C A125090 The characteristic polynomial of the n X n matrix has a root = 1+2*cos(2*Pi/(2n+1)).
%F A125090 f(n,x)=(x-1)f(n-1,x)-f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1.
%e A125090 Triangle starts:
%e A125090 1;
%e A125090 1, 0;
%e A125090 1, -1, -1;
%e A125090 1, -2, -1, 1;
%e A125090 1, -3, 0, 3, 0;
%e A125090 1, -4, 2, 5, -2, -1;
%e A125090 1, -5, 5, 6, -7, -2, 1;
%e A125090 1, -6, 9, 5, -15, 0, 5, 0;
%p A125090 with(linalg): m:=proc(i,j): if i=1 and j=1 then 0 elif i=j then 1 elif abs(i-j)=1 then 1 else 0 fi end: T:=proc(n,k) if n=0 and k=0 then 1 else coeff(charpoly(matrix(n,n,m),x),x,n-k) fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A125090 Cf. A104562.
%Y A125090 Sequence in context: A126886 A105685 A110858 this_sequence A073266 A125692 A128258
%Y A125090 Adjacent sequences: A125087 A125088 A125089 this_sequence A125091 A125092 A125093
%K A125090 sign,tabl
%O A125090 1,8
%A A125090 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 19 2006
%E A125090 Edited by njas, Nov 29 2006

%I A073266
%S A073266 1,1,1,0,2,1,1,1,3,1,0,2,3,4,1,0,2,4,6,5,1,0,0,6,8,10,6,1,1,1,3,13,15,
%T A073266 15,7,1,0,2,3,12,25,26,21,8,1,0,2,6,10,31,45,42,28,9,1,0,0,6,16,30,66,
%U A073266 77,64,36,10,1,0,2,4,18,40,76,126,126,93,45,11,1,0,0,6,16,50,96,168
%N A073266 Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.
%C A073266 Upper triangular region of the table A073265 read by rows. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 04 2005
%D A073266 S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
%F A073266 T(n, k)=coefficient of x^n in the formal power series (x+x^2+x^4+x^8+x^16+...)^k. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 04 2005
%F A073266 T(0, k) = T(n, 0) = 0, T(n, k) = 0 if k > n, T(n, 1) = 1 if n = 2^m, 0 otherwise, and in other cases T(n, k) = Sum_{i=0..[log2(n-1)]} T(n-(2^i), k-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 04 2005
%F A073266 Sum_{k, 0<=k<=n}T(n,k)=A023359(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 04 2006
%e A073266 T(6,3) = 4 because there are four ordered partitions of 6 into 3 powers of 2, namely: 4+1+1, 1+4+1, 1+1+4 and 2+2+2.
%e A073266 Triangle begins:
%e A073266 1;
%e A073266 1,1;
%e A073266 0,2,1;
%e A073266 1,1,3,1;
%e A073266 0,2,3,4,1;
%e A073266 0,2,4,6,5,1;
%Y A073266 Sequence in context: A105685 A110858 A125090 this_sequence A125692 A128258 A104967
%Y A073266 Adjacent sequences: A073263 A073264 A073265 this_sequence A073267 A073268 A073269
%K A073266 nonn,tabl
%O A073266 1,5
%A A073266 Antti Karttunen Jun 25 2002

%I A125692
%S A125692 1,1,1,0,2,1,1,1,3,1,0,2,3,4,1,2,2,2,6,5,1,0,4,6,0,10,6,1,5,5,3,11,5,15,
%T A125692 7,1,0,10,15,4,15,14,21,8,1,14,14,6,26,19,15,28,28,9,1,0,28,42,16,30,42,
%U A125692 7,48,36,10,1
%V A125692 1,-1,1,0,-2,1,1,1,-3,1,0,2,3,-4,1,-2,-2,2,6,-5,1,0,-4,-6,0,10,-6,1,5,5,-3,-11,-5,15,
%W A125692 -7,1,0,10,15,4,-15,-14,21,-8,1,-14,-14,6,26,19,-15,-28,28,-9,1,0,-28,-42,-16,30,42,-7,
%X A125692 -48,36,-10,1
%N A125692 Riordan array (1-x*c(-x^2),x(1-x*c(-x^2)) where c(x) is the g.f. of A000108.
%C A125692 Row sums are aerated signed Catalan numbers with g.f. c(-x^2). Inverse of A105306. First column is A105523.
%e A125692 Triangle begins
%e A125692 1,
%e A125692 -1, 1,
%e A125692 0, -2, 1,
%e A125692 1, 1, -3, 1,
%e A125692 0, 2, 3, -4, 1,
%e A125692 -2, -2, 2, 6, -5, 1,
%e A125692 0, -4, -6, 0, 10, -6, 1
%Y A125692 Sequence in context: A110858 A125090 A073266 this_sequence A128258 A104967 A098495
%Y A125692 Adjacent sequences: A125689 A125690 A125691 this_sequence A125693 A125694 A125695
%K A125692 easy,sign,tabl
%O A125692 0,5
%A A125692 Paul Barry (pbarry(AT)wit.ie), Nov 30 2006

%I A128258
%S A128258 1,0,1,1,2,1,1,1,3,1,1,0,0,4,1,0,3,1,0,5,1,1,0,0,0,0,6,1,0,0,3,1,0,0,7,
%T A128258 1,0,2,1,0,0,0,0,8,1,0,1,0,4,1,0,0,0,9,1
%V A128258 1,0,1,-1,2,1,-1,-1,3,1,-1,0,0,4,1,0,-3,-1,0,5,1,-1,0,0,0,0,6,1,0,0,-3,-1,0,0,7,1,0,-2,
%W A128258 -1,0,0,0,0,8,1,0,-1,0,-4,-1,0,0,0,9,1
%N A128258 Moebius transform of A128229.
%C A128258 Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...). A128259 = inverse Moebius transform of A128229.
%F A128258 A054525 * A128229
%e A128258 First few rows of the triangle are:
%e A128258 1;
%e A128258 0, 1;
%e A128258 -1, 2, 1;
%e A128258 -1, -1, 3, 1;
%e A128258 -1, 0, 0, 4, 1;
%e A128258 0, -3, -1, 0, 5, 1;
%e A128258 ...
%Y A128258 Cf. A128259, A128229, A054525, A000010.
%Y A128258 Sequence in context: A125090 A073266 A125692 this_sequence A104967 A098495 A095025
%Y A128258 Adjacent sequences: A128255 A128256 A128257 this_sequence A128259 A128260 A128261
%K A128258 tabl,sign
%O A128258 1,5
%A A128258 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 21 2007

%I A104967
%S A104967 1,1,1,1,2,1,1,1,3,1,1,0,0,4,1,1,1,2,2,5,1,1,2,3,4,5,6,1,1,3,3,3,5,9,7,
%T A104967 1,1,4,2,0,0,4,14,8,1,1,5,0,4,6,6,0,20,9,1,1,6,3,8,10,12,14,8,27,10,1,1,
%U A104967 7,7,11,10,10,14,22,21,35,11,1,1,8,12,12,5,0,0,8,27,40,44,12,1
%V A104967 1,-1,1,-1,-2,1,-1,-1,-3,1,-1,0,0,-4,1,-1,1,2,2,-5,1,-1,2,3,4,5,-6,1,-1,3,3,3,5,9,-7,1,
%W A104967 -1,4,2,0,0,4,14,-8,1,-1,5,0,-4,-6,-6,0,20,-9,1,-1,6,-3,-8,-10,-12,-14,-8,27,-10,1,-1,
%X A104967 7,-7,-11,-10,-10,-14,-22,-21,35,-11,1,-1,8,-12,-12,-5,0,0,-8,-27,-40,44,-12,1
%N A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.
%C A104967 Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.
%F A104967 G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
%e A104967 Triangle begins:
%e A104967 1;
%e A104967 -1,1;
%e A104967 -1,-2,1;
%e A104967 -1,-1,-3,1;
%e A104967 -1,0,0,-4,1;
%e A104967 -1,1,2,2,-5,1;
%e A104967 -1,2,3,4,5,-6,1;
%e A104967 -1,3,3,3,5,9,-7,1;
%e A104967 -1,4,2,0,0,4,14,-8,1;
%e A104967 -1,5,0,-4,-6,-6,0,20,-9,1; ...
%o A104967 (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)),n,x),k,y)}
%Y A104967 Cf. A090132, A104969, A104969.
%Y A104967 Sequence in context: A073266 A125692 A128258 this_sequence A098495 A095025 A069897
%Y A104967 Adjacent sequences: A104964 A104965 A104966 this_sequence A104968 A104969 A104970
%K A104967 sign,tabl
%O A104967 0,5
%A A104967 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005

%I A098495
%S A098495 1,1,0,1,1,1,1,2,1,1,1,3,1,1,0,1,4,5,1,1,1,1,5,11,7,2,1,1,1,6,19,29,9,1,
%T A098495 1,0,1,7,29,71,76,11,1,1,1,1,8,41,139,265,199,13,2,1,1,1,9,55,239,666,
%U A098495 989,521,15,1,1,0,1,10,71,377,1393,3191,3691,1364,17,1,1,1,1,11,89,559
%V A098495 1,1,0,1,-1,-1,1,-2,-1,-1,1,-3,1,1,0,1,-4,5,1,1,1,1,-5,11,-7,-2,-1,1,1,-6,19,-29,9,1,
%W A098495 -1,0,1,-7,29,-71,76,-11,1,1,-1,1,-8,41,-139,265,-199,13,-2,1,-1,1,-9,55,-239,666,-989,
%X A098495 521,-15,1,-1,0,1,-10,71,-377,1393,-3191,3691,-1364,17,1,-1,1,1,-11,89,-559
%N A098495 Array T(r,c) read by antidiagonals: values of triangle A098493 interpreted as polynomials, r>=0.
%D A098495 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%F A098495 Recurrence: T(r, 1) = 1, T(r, 2) = -r-1, T(r, c) = -rT(r, c-1) - T(r, c-2). (Corrected Oct 19 2004)
%e A098495 1,0,-1,-1,0,1,1,0,-1,
%e A098495 1,-1,-1,1,1,-1,-1,1,1,
%e A098495 1,-2,1,1,-2,1,1,-2,1,
%e A098495 1,-3,5,-7,9,-11,13,-15,
%e A098495 1,-4,11,-29,76,-199,521,
%e A098495 1,-5,19,-71,265,-989,3691,
%t A098495 T[r_, 1] := 1; T[r_, 2] := -r - 1; T[r_, c_] := -r*T[r, c - 1] - T[r, c - 2]; Flatten[ Table[ T[n - i, i], {n, 0, 11}, {i, n + 1}]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2005)
%o A098495 (PARI) { t(r,c)=if(c>r||c<0||r<0,0,if(c>=r-1,(-1)^r*if(c==r,1,-c),if(r==1,0,if(c==0,t(r-1,0)-t(r-2,0),t(r-1,c)-t(r-2,c)-t(r-1,c-1))))) } T(r,c)=sum(i=0,c,t(c,i)*r^i)
%Y A098495 See A094954 (with negative k) for negative r and more formulae and programs.
%Y A098495 Rows include (-1)^c times A005408, A002878, A001834, A030221, A002315. Columns include A028387. Antidiagonal sums are in A098496.
%Y A098495 Sequence in context: A125692 A128258 A104967 this_sequence A095025 A069897 A107682
%Y A098495 Adjacent sequences: A098492 A098493 A098494 this_sequence A098496 A098497 A098498
%K A098495 sign,tabl
%O A098495 0,8
%A A098495 Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 12 2004
%E A098495 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2005

%I A095025
%S A095025 1,1,2,1,1,1,3,1,1,1,1,1,2,0,2,1,0,1,2,0,1,1,1,1,0,2,1,1,3,1,3,0,1,0,0,
%T A095025 1,1,4,1,1,0,1,0,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,6,0,2,0,0,1,1,0,1,1,
%U A095025 1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,1,1,0
%N A095025 Number of cyclic difference sets with n elements.
%C A095025 A (v,k,lambda) cyclic difference set is a subset D={d_1,d_2,...,d_k} of the integers modulo v such that {1,2,...,v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways.
%H A095025 Dan Gordon, <a href="http://www.ccrwest.org/diffsets/diff_sets/">La Jolla Difference Set Repository</a>
%H A095025 Len Baumert and Dan Gordon, <a href="http://www.ccrwest.org/diffsets/papers/">Papers on Difference Sets</a>
%H A095025 Dan Gordon, <a href="http://www.ccrwest.org/diffsets/ds_list.pdf">List of Cyclic Difference Sets</a>
%e A095025 a(3)=1 corresponds to the (7,3,1) set {1,2,4}, a(4)=1 corresponds to the (14,4,1) set {0,1,3,9}.
%e A095025 a(5)=2 because there are two cyclic difference sets of length 5: The (v,k,lambda)=(11,5,2) set A095028={1,3,4,5,9} and the (21,5,1) set A095029= {3,6,7,12,14}
%Y A095025 Cf. A095029-A095047 examples of cyclic difference set with k=5..20.
%Y A095025 Sequence in context: A128258 A104967 A098495 this_sequence A069897 A107682 A085476
%Y A095025 Adjacent sequences: A095022 A095023 A095024 this_sequence A095026 A095027 A095028
%K A095025 nonn
%O A095025 3,3
%A A095025 Hugo Pfoertner (hugo(AT)pfoertner.org), May 27 2004

%I A069897
%S A069897 1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,1,2,2,5,1,1,1,6,1,3,1,2,1,1,3,8,1,1,
%T A069897 1,9,4,2,1,3,1,5,1,11,1,1,1,2,5,6,1,1,2,3,6,14,1,2,1,15,2,1,2,5,1,8,7,
%U A069897 3,1,1,1,18,1,9,1,6,1,2,1,20,1,3,3,21,9,5,1,2,1,11,10,23,3,1,1,3,3,2,1
%N A069897 Integer quotient of largest and smallest prime factors of n.
%F A069897 a(n)=Floor[A006530(n)/A020639(n)]
%Y A069897 Cf. A006530, A020639, A046665.
%Y A069897 Sequence in context: A104967 A098495 A095025 this_sequence A107682 A085476 A124944
%Y A069897 Adjacent sequences: A069894 A069895 A069896 this_sequence A069898 A069899 A069900
%K A069897 nonn
%O A069897 2,9
%A A069897 Labos E. (labos(AT)ana.sote.hu), Apr 10 2002

%I A107682
%S A107682 0,1,1,2,1,1,1,3,1,1,1,1,4,1,1,1,1,1,5,1,1,1,1,1,1,6,1,1,1,1,1,1,1,7,1,
%T A107682 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,9,9,2,2,2,3,3,2,2,2,2,4,4,2,2,2,2,2,
%U A107682 5,5,2,2,2,2,2,2,6,6,2,2,2,2,2,2,2,7,7,2,2,2,2,2,2,2,2,8,8,2,2,2,2,2,2
%N A107682 The size of every run of identical digits is given by the digit immediately following immediately the said run. There can be no two identical runs in the sequence.
%C A107682 The sequence is finite.
%H A107682 E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/AutoChunks.htm">Auto-chunks</a>.
%e A107682 First run ("0") is made of 1 "identical digit(s)", thus is followed by 1; second run ("11") is made of 2 identical digits, thus followed by 2; third run ("2") is made of 1 "identical digit(s)", thus followed by 1; etc.
%Y A107682 Sequence in context: A098495 A095025 A069897 this_sequence A085476 A124944 A094392
%Y A107682 Adjacent sequences: A107679 A107680 A107681 this_sequence A107683 A107684 A107685
%K A107682 base,easy,fini,nonn
%O A107682 0,4
%A A107682 Eric Angelini (eric.angelini(AT)kntv.be), Jun 10 2005

%I A085476
%S A085476 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,3,1,1,1,1,5,6,1,2,1,1,1,6,10,4,1,1,1,
%T A085476 1,1,7,15,10,1,3,1,1,1,1,8,21,20,5,1,3,2,1,1,1,9,28,35,15,1,4,1,1,1,1,1,
%U A085476 10,36,56,35,6,1,6,1,1,1,1
%N A085476 Periodic Pascal array, read by antidiagonals.
%C A085476 G.f. of binomial transform of k-th row is given by 1/((1-x)^(k+1)-x^(k+1))
%F A085476 Square array T(n, k)=C(k, n mod (k+1)).
%e A085476 Rows begin
%e A085476 1 1 1 1 1 1 ...
%e A085476 1 1 1 1 1 1 ...
%e A085476 1 2 1 1 2 1 ...
%e A085476 1 3 3 1 1 3 ...
%e A085476 1 4 6 4 1 1 ...
%Y A085476 Cf. A007318, A000749, A049016, A006090, A049017.
%Y A085476 Sequence in context: A095025 A069897 A107682 this_sequence A124944 A094392 A111946
%Y A085476 Adjacent sequences: A085473 A085474 A085475 this_sequence A085477 A085478 A085479
%K A085476 easy,nonn,tabl
%O A085476 0,8
%A A085476 Paul Barry (pbarry(AT)wit.ie), Jul 02 2003

%I A124944
%S A124944 1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,3,1,1,1,1,6,4,1,1,1,1,1,8,6,3,1,1,1,1,
%T A124944 1,11,8,5,1,1,1,1,1,1,15,11,7,3,1,1,1,1,1,1,20,15,9,5,1,1,1,1,1,1,1,26,
%U A124944 21,12,8,3,1,1,1,1,1,1,1,35,27,16,10,5,1,1,1,1,1,1,1,1,45,37,21,13,8,3
%N A124944 Table, number of partitions of n with k as high median.
%C A124944 For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
%e A124944 For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
%Y A124944 Cf. A124943, A026794, A008284, A000041 (row sums), A027336 (1st column).
%Y A124944 Sequence in context: A069897 A107682 A085476 this_sequence A094392 A111946 A137844
%Y A124944 Adjacent sequences: A124941 A124942 A124943 this_sequence A124945 A124946 A124947
%K A124944 nonn,tabl
%O A124944 1,7
%A A124944 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 13 2006

%I A094392
%S A094392 1,1,1,1,1,2,1,1,1,3,1,1,1,1,5,1,1,1,1,2,8,1,1,1,1,1,3,13,1,1,1,1,1,1,5,
%T A094392 21,1,1,1,1,1,1,2,7,34,1,1,1,1,1,1,1,3,11,55,1,1,1,1,1,1,1,1,5,16,891,1,
%U A094392 1,1,1,1,1,1,2,7,25,144,1,1,1,1,1,1,1,1,1,3,11,37,233,1,1,1,1,1,1,1,1,1
%N A094392 Antidiagonals of the tables formed from b(m,2,n,n), which is defined in the reference.
%C A094392 This sequence can be used to help find an extension for A006209.
%D A094392 B.-S. Du, A simple method which generates infinitely many congruence indentities, Fib. Quart., 27 (1989), 116-124.
%F A094392 For i=2 and k >= 1 b(k+2, 2, n, n)=b(k, 2, 1, n) + b(k+1, 2, n, n). The remaining portion for the recurrence is defined in the reference.
%e A094392 E.g. for m = 5 and n = 2, b(5,2,2,2)= b(3,2,1,2) + b(4,2,2,2)= 2 because of the definition in the reference.
%e A094392 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 5 2 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 8 3 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 13 5 2 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 21 7 3 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 34 11 5 2 1 1 1 1 1 1 1 1 1 1 1
%e A094392 55 16 7 3 1 1 1 1 1 1 1 1 1 1 1
%e A094392 89 25 11 5 2 1 1 1 1 1 1 1 1 1 1
%e A094392 144 37 15 7 3 1 1 1 1 1 1 1 1 1 1
%e A094392 233 57 23 11 5 2 1 1 1 1 1 1 1 1 1
%e A094392 377 85 32 15 7 3 1 1 1 1 1 1 1 1 1
%e A094392 610 130 49 23 11 5 2 1 1 1 1 1 1 1 1
%p A094392 b := proc(k,i,j,n) option remember; if k = 1 then if i = 1 then return 0; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if k = 2 then if i = 1 then return 1; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if j = n then return b(k-2, i, 1, n) + b(k-1, i, n, n); end if; return b(k-2, i, 1, n) + b(k-2, i, j+1, n); end proc; (Deugau)
%Y A094392 Cf. A006209.
%Y A094392 Sequence in context: A107682 A085476 A124944 this_sequence A111946 A137844 A079229
%Y A094392 Adjacent sequences: A094389 A094390 A094391 this_sequence A094393 A094394 A094395
%K A094392 nonn,tabl
%O A094392 1,6
%A A094392 Amy Robinson (amylou(AT)mchsi.com), Apr 28 2004
%E A094392 Corrected and extended by Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005

%I A111946
%S A111946 1,1,1,1,1,2,1,1,1,3,1,1,1,1,5,1,1,2,1,1,8,1,1,1,1,1,1,13,1,1,1,3,1,1,1,
%T A111946 21,1,1,2,1,1,2,1,1,34,1,1,1,1,5,1,1,1,1,55,1,1,1,1,1,1,1,1,1,1,89,1,1,
%U A111946 2,3,1,8,1,3,2,1,1,144,1,1,1,1,1,1,1,1,1,1,1,1,233,1,1,1,1,1,1,13,1,1,1
%N A111946 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.
%D A111946 P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.
%F A111946 T(n, k) = Fibonacci(g), where g = gcd(n, k).
%Y A111946 Cf. A000045, A111956, A111957.
%Y A111946 Sequence in context: A085476 A124944 A094392 this_sequence A137844 A079229 A115621
%Y A111946 Adjacent sequences: A111943 A111944 A111945 this_sequence A111947 A111948 A111949
%K A111946 nonn,tabl
%O A111946 1,6
%A A111946 njas, Nov 28 2005

%I A137844
%S A137844 1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,1,2,
%T A137844 1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,6,1,1,1,2,1,1,1,3,
%U A137844 1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,1,2,1,1,1,3,1,1,1,2
%N A137844 Define S(1) = {1}, S(n+1) = S(n) U S(n) if a(n) is even, S(n+1) = S(n) U n U S(n) if a(n) is odd. Sequence {a(n), n >= 1} is limit as n approaches infinity of S(n). (U represents concatenation.).
%e A137844 S(1) = {1}.
%e A137844 S(2) = {1,1,1}, because a(1) = 1, which is odd.
%e A137844 S(3) = {1,1,1,2,1,1,1}, because a(2) = 1, which is odd.
%e A137844 S(4) = {1,1,1,2,1,1,1,3,1,1,1,2,1,1,1}.
%e A137844 S(5) = {1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1}, because a(4) = 2, which is even.
%e A137844 Etc.
%Y A137844 Cf. A137843.
%Y A137844 Sequence in context: A124944 A094392 A111946 this_sequence A079229 A115621 A077565
%Y A137844 Adjacent sequences: A137841 A137842 A137843 this_sequence A137845 A137846 A137847
%K A137844 easy,nonn
%O A137844 1,4
%A A137844 Leroy Quet (qq-quet(AT)mindspring.com), Feb 13 2008

%I A079229
%S A079229 1,1,2,1,1,1,3,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,2,1,3,1,1,1,1,2,1,1,1,2,
%T A079229 1,1,1,2,1,1,1,3,2,1,1,4,1,1,1,2,1,2,1,2,1,1,1,2,1,1,3,2,1,1,1,2,1,1,1,
%U A079229 2,1,1,3,1,1,1,1,3,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,4,1,2,1,1,1,2,1,1
%N A079229 Least k>0 such that sfk(n+k) > sfk(n), where sfk is the square-free kernel (A007947).
%Y A079229 a(n) = A079228(n) - n.
%Y A079229 Sequence in context: A094392 A111946 A137844 this_sequence A115621 A077565 A115561
%Y A079229 Adjacent sequences: A079226 A079227 A079228 this_sequence A079230 A079231 A079232
%K A079229 nonn
%O A079229 1,3
%A A079229 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 02 2003

%I A115621
%S A115621 1,1,2,1,1,1,3,1,1,1,2,1,2,4,1,1,1,1,1,1,2,1,2,1,3,5,1,1,1,1,1,2,1,2,1,
%T A115621 1,1,3,1,3,2,2,1,4,6,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,3,1,1,2,1,3,1,4,
%U A115621 2,3,1,5,7,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,2,1,3,1,1,2,2,2,1,1,2
%N A115621 Signature of partitions in Abramowitz and Stegun order.
%C A115621 The signature of a partition is a partition consisting of the repetition factors of the original partition. E.g. [1,1,3,4,4] = [1^2,3^1,4^2], so the repetition factors are 2,1,2, making the signature [1,2,2] = [1,2^2].
%C A115621 The sum (or order) of the signature is the number of parts of the original partition, and the number of parts of the signature is the number of distinct parts of the original partition.
%H A115621 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%e A115621 [1]; [1], [2]; [1], [1,1], [3]; [1], [1,1], [2], [1,2], [4]; ...
%Y A115621 Cf. A036036, A113787, A115622, part counts A103921, row counts A000070.
%Y A115621 Sequence in context: A111946 A137844 A079229 this_sequence A077565 A115561 A115622
%Y A115621 Adjacent sequences: A115618 A115619 A115620 this_sequence A115622 A115623 A115624
%K A115621 nonn,tabf
%O A115621 1,3
%A A115621 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2006

%I A077565
%S A077565 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,3,1,3,1,1,1,4,1,1,2,3,1,4,1,3,1,1,1,
%T A077565 3,1,1,1,4,1,4,1,3,3,1,1,6,1,3,1,3,1,4,1,4,1,1,1,5,1,1,3,4,1,4,1,3,1,4,
%U A077565 1,7,1,1,3,3,1,4,1,6,2,1,1,5,1,1,1,4,1,5,1,3,1,1,1,9,1,3,3,3,1,4,1,4,4
%N A077565 Number of factorizations ( Smarandache Factor Partitions ) into parts with distinct prime signature.
%D A077565 Amarnath Murthy, Generalization of partition function. Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 1-2-3,2000.
%e A077565 a(24) = 4, 24 = 12*2 = 8*3 = 6*4. The partitions 2*3*4, 2*2*2*3 etc. are not counted.
%Y A077565 Cf. A077563, A077564, A077566.
%Y A077565 Sequence in context: A137844 A079229 A115621 this_sequence A115561 A115622 A108886
%Y A077565 Adjacent sequences: A077562 A077563 A077564 this_sequence A077566 A077567 A077568
%K A077565 nonn
%O A077565 1,8
%A A077565 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 11 2002
%E A077565 Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Aug 26 2003

%I A115561
%S A115561 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,3,1,5,1,1,1,2,1,1,3,7,1,5,1,2,1,1,1,
%T A115561 3,1,1,1,2,1,7,1,11,5,1,1,2,1,5,1,13,1,3,1,2,1,1,1,3,1,1,7,2,1,11,1,17,
%U A115561 1,7,1,2,1,1,5,19,1,13,1,2,3,1,1,3,1,1,1,2,1,3,1,23,1,1,1,2,1,7,11,5,1
%N A115561 lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.
%C A115561 a(n) = A020639(A054576(n)).
%H A115561 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeastPrimeFactor.html">Least Prime Factor</a>
%Y A115561 Cf. A032742, A014673, A117358.
%Y A115561 Sequence in context: A079229 A115621 A077565 this_sequence A115622 A108886 A001492
%Y A115561 Adjacent sequences: A115558 A115559 A115560 this_sequence A115562 A115563 A115564
%K A115561 nonn
%O A115561 1,8
%A A115561 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 10 2006

%I A115622
%S A115622 1,1,2,1,1,1,3,1,1,1,2,2,1,4,1,1,1,1,1,2,1,2,1,3,1,5,1,1,1,1,1,2,1,2,1,
%T A115622 1,1,3,1,3,2,2,4,1,6,1,1,1,1,1,2,1,1,1,1,1,1,3,1,2,1,2,1,2,1,1,4,1,3,1,
%U A115622 3,2,5,1,7,1,1,1,1,1,2,1,1,1,1,1,1,3,1,2,1,1,1,2,1,2,1,1,4,1,2,1,2,2,2
%N A115622 Signature of partitions in Mathematica order.
%C A115622 The signature of a partition is a partition consisting of the repetition factors of the original partition. E.g. [4,4,3,1,1] = [4^2,3^1,1^2], so the repetition factors are 2,1,2, making the signature [2,2,1] = [2^2,1].
%C A115622 The sum (or order) of the signature is the number of parts of the original partition, and the number of parts of the signature is the number of distinct parts of the original partition.
%e A115622 [1]; [1], [2]; [1], [1,1], [3]; [1], [1,1], [2], [2,1], [4]; ...
%Y A115622 Cf. A080577, A115624, A115621, part counts A115623, row counts A000070.
%Y A115622 Sequence in context: A115621 A077565 A115561 this_sequence A108886 A001492 A054576
%Y A115622 Adjacent sequences: A115619 A115620 A115621 this_sequence A115623 A115624 A115625
%K A115622 nonn,tabf
%O A115622 1,3
%A A115622 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2006

%I A108886
%S A108886 1,1,1,1,2,1,1,1,3,1,1,1,3,4,1,1,1,5,2,5,1,1,1,1,20,5,6,1,1,1,1,35,35,3,
%T A108886 7,1,1,1,1,1,1,28,7,8,1,1,1,1,1,9,63,42,4,9,1,1,1,1,1,15,42,35,120,9,10,
%U A108886 1,1,1,1,1,1,231,11,66,55,5,11,1,1,1,1,1,1,396,231,72,45,55,11,12,1,1,1
%N A108886 Let T(m,p) be the value of the following game: there are m ``minus'' balls and p ``plus'' balls in an urn, for a total of n=m+p balls. You may draw balls from the urn one at a time at random and without replacement until you decide to stop drawing. Each minus ball drawn costs you $1 and each plus ball drawn gets you $1. Sequence gives triangle of denominators of T(n-p,p), 0 <= p <= n, read by rows.
%D A108886 L. A. Shepp, Stochastic Processes [Course], Statistics Dept., Rutgers University, 2004.
%F A108886 T(m, 0)=0, T(0, p)=p; T(m, p) = max{0, (m/(m+p))*(-1+T(m-1, p))+(p/(m+p))*(1+T(m, p-1))}.
%e A108886 Triangle of values T(n-p,p), 0 <= p <= n, begins:
%e A108886 [0]
%e A108886 [0, 1]
%e A108886 [0, 1/2, 2]
%e A108886 [0, 0, 4/3, 3]
%e A108886 [0, 0, 2/3, 9/4, 4]
%e A108886 [0, 0, 1/5, 3/2, 16/5, 5]
%e A108886 [0, 0, 0, 17/20, 12/5, 25/6, 6]
%e A108886 [0, 0, 0, 12/35, 58/35, 10/3, 36/7, 7]
%e A108886 [0, 0, 0, 0, 1, 71/28, 30/7, 49/8, 8]
%p A108886 M:=60; for m from 0 to M do T(m,0):=0; od: for p from 0 to M do T(0,p):=p; od: for n from 1 to M do for m from 1 to n-1 do p:=n-m; t1:=(m/(m+p))*(-1+T(m-1,p))+(p/(m+p))*(1+T(m,p-1)); T(m,p):=max(0,t1); od: od:
%Y A108886 Cf. A108885. Sequence T(m,m) is A108883/A108884.
%Y A108886 Sequence in context: A077565 A115561 A115622 this_sequence A001492 A054576 A138904
%Y A108886 Adjacent sequences: A108883 A108884 A108885 this_sequence A108887 A108888 A108889
%K A108886 nonn,tabl,frac
%O A108886 0,5
%A A108886 njas, Jul 16 2005

%I A001492
%S A001492 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,1,1,1,5,1,1,1,6,1,1,1,7,1,1,1,8,
%T A001492 1,1,1,9,1,1,1,10,1,1,1,11,1,1,1,12,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,
%U A001492 1,1,1,5,1,1,1,6,1,1,1,7,1,1,1,8,1,1,1,9
%N A001492 Clock chimes with a quarter-hour bell.
%o A001492 (PARI) a(n)=if((n+1)%4,1,((n+1)/4-1)%12+1)
%Y A001492 Cf. A007879, A007884.
%Y A001492 Sequence in context: A115561 A115622 A108886 this_sequence A054576 A138904 A135222
%Y A001492 Adjacent sequences: A001489 A001490 A001491 this_sequence A001493 A001494 A001495
%K A001492 nonn,easy
%O A001492 0,8
%A A001492 njas

%I A054576
%S A054576 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,1,3,1,5,1,1,1,6,1,1,3,7,1,5,1,8,1,1,1,
%T A054576 9,1,1,1,10,1,7,1,11,5,1,1,12,1,5,1,13,1,9,1,14,1,1,1,15,1,1,7,16,1,11,
%U A054576 1,17,1,7,1,18,1,1,5,19,1,13,1,20,9,1,1,21,1,1,1,22,1,15,1,23,1,1,1,24
%N A054576 Largest proper factor of largest proper factor of n.
%F A054576 a(n) = A053598(A053598(n))
%F A054576 a(n)=A032742(A032742(n)); A117357(n)=A020639(a(n)); A117358(n)=A032742(a(n))=a(n)/A117357(n); a(A037143(n))=1, a(A033942(n))>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 10 2006
%Y A054576 Cf. A053598.
%Y A054576 Sequence in context: A115622 A108886 A001492 this_sequence A138904 A135222 A124094
%Y A054576 Adjacent sequences: A054573 A054574 A054575 this_sequence A054577 A054578 A054579
%K A054576 easy,nonn
%O A054576 1,8
%A A054576 Henry Bottomley (se16(AT)btinternet.com), Apr 11 2000

%I A138904
%S A138904 1,1,1,2,1,1,1,3,1,1,2,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,
%T A138904 1,2,1,1,1,1,1,3,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,
%U A138904 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A138904 Number of rotational symmetries in the binary expansion of a number.
%C A138904 Mersenne numbers of form (2^n - 1) have n rotational symmetries.
%C A138904 For prime length binary expansions these are the only nontrivial symmetries.
%C A138904 For composite length expansions it seems that when the number of symmetries is non-trivial it is equal to a factor of the length. We're working on an explicit formula.
%C A138904 Discovered in the context of random circulant matricies, examining if there's a correlation between degrees of freedom and number of symmetries in the first row.
%C A138904 When combined with A138954, these two sequences should give a full account of the number of redundant rows in a circulant square matrix with at most two distinct values, where a(n) is the encoding of the first row of the matrix into binary such that value a = 1, and value b = 0.
%C A138904 Discovered on the night of Apr 02, 2008 by Maxwell Sills and Gary Doran.
%C A138904 Conjecture: For binary expansions of length n, there are d(n) distinct values that will show up as symmetries, where d is the divisor function. The symmetry values will be precisely the divisors of n.
%C A138904 Example: for binary expansions of length 12, one sees that d(12) = 6 distinct values show up as symmetries (1, 2, 3, 4, 6, 12).
%C A138904 Conjecture: For numbers whose binary expansion has length n which has proper divisors which are all coprime: There will be only one number of length n with n symmetires. That number is 2^n - 1. For each proper divisor d (excluding 1), you can generate all numbers of length n that have n/d symmetries like so: (2^0 + 2^d + 2^2d ... 2^(n-d)) * a, where 2^(d-1) <= a < (2^d) - 1. The rest of the expansions of length n will have only the trivial symmetry.
%H A138904 Maxwell Sills and Gary Doran, <a href="http://www.research.att.com/~njas/sequences/b138904.txt">Table of n, a(n) for n = 0..99</a>
%e A138904 a(10) = 2 because the binary expansion of 10 is 1010 and it has two rotational symmetries (including identity).
%Y A138904 Cf. A136441, A138954.
%Y A138904 Sequence in context: A108886 A001492 A054576 this_sequence A135222 A124094 A101950
%Y A138904 Adjacent sequences: A138901 A138902 A138903 this_sequence A138905 A138906 A138907
%K A138904 base,easy,nonn
%O A138904 0,4
%A A138904 Max Sills (maxwell.sills(AT)case.edu), Apr 03 2008, Apr 04 2008

%I A135222
%S A135222 1,1,1,2,1,1,1,3,1,1,2,1,4,1,1,1,4,1,5,1,1,2,1,7,1,6,1,1,1,5,1,11,1,7,1,
%T A135222 1,2,1,11,1,16,1,8,1,1,1,6,1,21,1,22,1,9,1,1
%N A135222 A049310 + A000012 - I.
%C A135222 Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28,...).
%F A135222 A049310 + A000012 - Identity matrix, as infinite lower triangular matrices.
%e A135222 First few rows of the triangle are:
%e A135222 1;
%e A135222 1, 1;
%e A135222 2, 1, 1;
%e A135222 1, 3, 1, 1;
%e A135222 2, 1, 4, 1, 1;
%e A135222 1, 4, 1, 5, 1, 1;
%e A135222 2, 1, 7, 1, 6, 1, 1;
%e A135222 1, 5, 1, 11, 1, 7, 1, 1;
%e A135222 2, 1, 11, 1, 16, 1, 8, 1, 1;
%e A135222 ...
%Y A135222 Cf. A049310, A081659.
%Y A135222 Sequence in context: A001492 A054576 A138904 this_sequence A124094 A101950 A104562
%Y A135222 Adjacent sequences: A135219 A135220 A135221 this_sequence A135223 A135224 A135225
%K A135222 nonn,tabl
%O A135222 1,4
%A A135222 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007

%I A124094
%S A124094 1,1,1,1,1,2,1,1,1,3,1,1,2,2,5,1,1,1,2,2,7,1,1,2,2,4,3,11,1,1,1,3,2,5,4,
%T A124094 15,1,1,2,1,5,3,7,5,22,1,1,1,3,1,6,4,9,6,30,1,1,2,2,5,2,10,5,13,8,42,1,
%U A124094 1,1,2,2,7,2,13,6,16,10,56,1,1,2,2,4,3,11,3,19,8,22,12,77,1,1,1,3,2,5,4
%N A124094 Table T(n,m) giving number of partitions of n such that all parts are coprime to m. Read along antidiagonals (increasing n, decreasing m).
%H A124094 N. Robbins, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Robbins/robbins4.html">On partition functions and divisor sums</a>, J. Int. Sequences, 5 (2002) 02.1.4.
%e A124094 Upper left corner of table starts with row m=1 and column n=0:
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243,297,355,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,3,5,6,10,13,19,25,34,44,60, 76,100,127,164,205,262,325,409,505,628,769,
%e A124094 1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23,26,
%e A124094 1,1,2,3,5,7,11,14,21,28,39,51,70, 90,119,153,199,252,324,406,515,642,804,994,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243,297,355,
%e A124094 1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41, 47,54,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,55,76, 99,132,171,224,286,370,468,597,750,945,1177,
%e A124094 1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23,26,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,100,134,174,228,292,378,479,612,770,972,1213,
%e A124094 1,1,1,2,2,3, 4, 4, 5, 7, 8,10,12, 14, 17, 21, 24, 28, 34, 39, 46, 53, 61,71,
%e A124094 1,1,2,2,4,4, 6, 7,11,12,16,19,25, 29, 37, 44, 56, 65, 80, 94,114,133,160,187,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,296,384,488,624,787,995,1244,
%e A124094 1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23,26,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,489,626,790,999,1250,
%e A124094 1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41, 47,54,
%o A124094 (PARI) sigmastar(n,m)= { local(d,res=0) ; d=divisors(n) ; for(i=1,matsize(d)[2], if( gcd(d[i],m)==1, res += d[i] ; ) ; ) ; return(res) ; } f(n,m)= { local(qvec=vector(n+1,i,gcd(1,m))) ; for(i=1,n, qvec[i+1]=sum(k=1,i,sigmastar(k,m)*qvec[i-k+1])/i ; ) ; return(qvec[n+1]) ; } { for(d=1,18, for(c=0,d-1, r=d-c ; print1(f(c,r),",") ; ) ; ) ; }
%Y A124094 Row m=1 is A000041. Rows m=2,4,8,... (where m is a power of 2) are A000009. Rows m=3,9,... (where m is a power of 3) are A000726. Row m=5 is A035959. Row=7 is A035985. Row m=10 is A096938.
%Y A124094 Sequence in context: A054576 A138904 A135222 this_sequence A101950 A104562 A111603
%Y A124094 Adjacent sequences: A124091 A124092 A124093 this_sequence A124095 A124096 A124097
%K A124094 easy,nonn,tabl
%O A124094 0,6
%A A124094 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 26 2006

%I A101950
%S A101950 1,1,1,0,2,1,1,1,3,1,1,2,3,4,1,0,4,2,6,5,1,1,2,9,0,10,6,1,1,3,9,15,5,15,
%T A101950 7,1,0,6,3,24,20,14,21,8,1,1,3,18,6,49,21,28,28,9,1,1,4,18,36,35,84,14,
%U A101950 48,36,10,1,0,8,4,60,50,98,126,6,75,45,11,1,1,4,30,20,145,36,210
%V A101950 1,1,1,0,2,1,-1,1,3,1,-1,-2,3,4,1,0,-4,-2,6,5,1,1,-2,-9,0,10,6,1,1,3,-9,-15,5,15,7,1,0,
%W A101950 6,3,-24,-20,14,21,8,1,-1,3,18,-6,-49,-21,28,28,9,1,-1,-4,18,36,-35,-84,-14,48,36,10,1,
%X A101950 0,-8,-4,60,50,-98,-126,6,75,45,11,1,1,-4,-30,20,145,36,-210
%N A101950 Product of A049310 and A007318 as lower triangular matrices.
%C A101950 A Chebyshev and Pascal product.
%C A101950 Row sums are n+1, diagonal sums the constant sequence 1. Riordan array (1/(1-x+x^2,x/(1-x+x^2)).
%C A101950 Apart from signs, identical with A104562.
%F A101950 Number triangle T(n, k) = sum{k=0..n, (-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}
%e A101950 Rows begin {1}, {1,1}, {0,2,1}, {-1,1,3,1}, { -1,-2,3,4,1},..
%Y A101950 Cf. A104562.
%Y A101950 Sequence in context: A138904 A135222 A124094 this_sequence A104562 A111603 A136178
%Y A101950 Adjacent sequences: A101947 A101948 A101949 this_sequence A101951 A101952 A101953
%K A101950 easy,sign,tabl
%O A101950 0,5
%A A101950 Paul Barry (pbarry(AT)wit.ie), Dec 22 2004

%I A104562
%S A104562 1,1,1,0,2,1,1,1,3,1,1,2,3,4,1,0,4,2,6,5,1,1,2,9,0,10,6,1,1,3,9,15,5,15,7,
%T A104562 1,0,6,3,24,20,14,21,8,1,1,3,18,6,49,21,28,28,9,1,1,4,18,36,35,84,14,48,
%U A104562 36,10,1,0,8,4,60,50,98,126,6,75,45,11,1,1,4,30,20,145,36,210,168,45,110,55
%V A104562 1,-1,1,0,-2,1,1,1,-3,1,-1,2,3,-4,1,0,-4,2,6,-5,1,1,2,-9,0,10,-6,1,-1,3,9,-15,-5,15,-7,
%W A104562 1,0,-6,3,24,-20,-14,21,-8,1,1,3,-18,-6,49,-21,-28,28,-9,1,-1,4,18,-36,-35,84,-14,-48,
%X A104562 36,-10,1,0,-8,4,60,-50,-98,126,6,-75,45,-11,1,1,4,-30,-20,145,-36,-210,168,45,-110,55
%N A104562 Inverse of the Motzkin triangle A064189.
%C A104562 Or, triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n tridiagonal matrix with 1's on the main, sub-, and superdiagonal (0<=k<=n). The characteristic polynomial has a root 1+2cos(Pi/(n+1)). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 19 2006
%C A104562 Row sums have g.f. 1/(1+x^2); diagonal sums are (-1)^n. Riordan array (1/(1+x+x^2), x/(1+x+x^2)).
%C A104562 Apart from signs, identical to A101950.
%C A104562 Or, triangle read by rows in which row n gives coefficients of characteristic polynomial of tridaigonal matrix with 1's on the main diagonal and -1's on the two adjacent diagonals. For example: M(3)={{1, -1, 0}, {-1, 1, -1}, {0, -1, 1}}. - from More terms from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2008
%D A104562 Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
%F A104562 Number triangle T(n, k)=sum{j=0..n, (-1)^(k-j)*(-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}
%e A104562 Triangle starts:
%e A104562 1;
%e A104562 -1,1;
%e A104562 0,-2,1;
%e A104562 1,1,-3,1;
%e A104562 -1,2,3,-4,1;
%e A104562 0,-4,2,6,-5,1;
%p A104562 with(linalg): m:=proc(i,j) if abs(i-j)<=1 then 1 else 0 fi end: T:=(n,k)->coeff(charpoly(matrix(n,n,m),x),x,k): 1; for n from 1 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A104562 a0[n_] := 1; b[n_] := -1; T[n_, m_, d_] := If[ n == m, a0[n], If[n == m - 1 || n == m + 1, If[n == m - 1, b[m - 1], If[n == m + 1, b[n - 1], 0]], 0]]; MO[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ MO[n], x], x], {n, 1, 10}]]; Flatten[a] - from More terms from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2008
%Y A104562 Cf. A125090, A101950.
%Y A104562 Sequence in context: A135222 A124094 A101950 this_sequence A111603 A136178 A047140
%Y A104562 Adjacent sequences: A104559 A104560 A104561 this_sequence A104563 A104564 A104565
%K A104562 easy,sign,tabl
%O A104562 0,5
%A A104562 Paul Barry (pbarry(AT)wit.ie), Mar 15 2005
%E A104562 Edited by njas, Apr 10 2008

%I A111603
%S A111603 1,1,1,1,2,1,1,1,3,1,1,2,3,4,1,1,2,1,2,5,1,1,2,3,4,5,6,1,1,1,3,3,5,3,7,
%T A111603 1,1,2,3,4,5,2,7,8,1,1,2,3,4,1,3,7,4,9,1,1,2,3,4,5,6,7,8,9,10,1,1,1,3,1,
%U A111603 5,6,7,2,3,5,11,1,1,2,3,4,5,6,7,8,9,10,11,12,1,1,1,3,4,5,3,1,4,9,10
%N A111603 Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from upper right to lower left.
%e A111603 Table begins
%e A111603 k= 0 1 2 3 4 5 6 7 8 9 10 11 12 13
%e A111603 n\
%e A111603 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A111603 2| 1 2 1 2 2 2 1 2 2 2 1 2 1 2
%e A111603 3| 1 3 3 1 3 3 3 3 3 3 3 3 1 3
%e A111603 4| 1 4 2 4 3 4 4 4 1 4 4 4 3 4
%e A111603 5| 1 5 5 5 5 1 5 5 5 5 4 5 5 5
%e A111603 6| 1 6 3 2 3 6 6 6 3 4 6 6 6 6
%e A111603 7| 1 7 7 7 7 7 7 1 7 7 7 7 7 7
%e A111603 8| 1 8 4 8 2 8 4 8 7 8 8 8 4 8
%e A111603 9| 1 9 9 3 9 9 3 9 9 1 9 9 6 9
%e A111603 10| 1 10 5 10 10 2 5 10 10 10 3 10 5 10
%e A111603 11| 1 11 11 11 11 11 11 11 11 11 11 1 11 11
%e A111603 12| 1 12 6 4 9 12 4 12 12 8 6 12 6 12
%e A111603 13| 1 13 13 13 13 13 13 13 13 13 13 13 13 1
%e A111603 14| 1 14 7 14 7 14 14 2 7 14 14 14 14 14
%e A111603 15| 1 15 15 5 15 3 10 15 15 10 15 15 5 15
%e A111603 16| 1 16 8 16 4 16 8 16 10 16 8 16 12 16
%t A111603 f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[ a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]]; Flatten[ Table[ f[i, n - i], {n, 15}, {i, n - 1, 1, -1}]]
%Y A111603 Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
%Y A111603 Cf. A109626, A111604.
%Y A111603 Sequence in context: A124094 A101950 A104562 this_sequence A136178 A047140 A047150
%Y A111603 Adjacent sequences: A111600 A111601 A111602 this_sequence A111604 A111605 A111606
%K A111603 nonn,tabl
%O A111603 1,5
%A A111603 Paul D. Hanna (pauldhanna(AT)juno.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 01 2005

%I A136178
%S A136178 1,1,1,2,1,1,1,3,1,1,2,4,1,3,1,1,5,1,1,1,1,2,6,1,1,1,7,1,1,5,1,2,4,8,1,
%T A136178 1,1,3,3,9,1,1,2,1,5,10
%N A136178 Irregular array read by rows: row n contains the GCDs of each pair of consecutive positive divisors of n.
%C A136178 Each row has d(n)-1 terms, where d(n) is the number of positive divisors of n. The first row listed is row 2.
%e A136178 The positive divisors of 20 are 1,2,4,5,10,20. GCD(1,2)=1. GCD(