A279041 - OEIS

%I #10 Dec 28 2018 19:26:28

%S 1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,

%T 7,7,7,8,8,8,10,10,11,11,11,12,12,12,14,14,15,15,15,16,16,16,18,18,19,

%U 19,19,21,21,22,24,25,26,26,26,28,28,29,31,32,33,33,33,35,35,36,39,40,42,42,43,45,46,47,50,51,53

%N Expansion of Product_{k>=1} 1/(1 - x^(k*(3*k-2))).

%C Number of partitions of n into nonzero octagonal numbers (A000567).

%H Alois P. Heinz, <a href="/A279041/b279041.txt">Table of n, a(n) for n = 0..20000</a>

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=1} 1/(1 - x^(k*(3*k-2))).

%e a(9) = 2 because we have [8, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

%p h:= proc(n) option remember; `if`(n<1, 0, (t->

%p       `if`(t*(3*t-2)>n, t-1, t))(1+h(n-1)))

%p     end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(3*i-2))))

%p     end:

%p a:= n-> b(n, h(n)):

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 28 2018

%t nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (3 k - 2))), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000567, A001156, A007294, A037444, A218379, A278949, A279012.

%K nonn,easy

%O 0,9

%A _Ilya Gutkovskiy_, Dec 04 2016