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%I #6 Feb 17 2017 09:49:28
%S 1,1,1,1,2,3,4,5,7,10,15,21,29,40,57,81,114,159,223,315,445,626,879,
%T 1236,1741,2452,3450,4852,6826,9608,13524,19032,26778,37680,53027,
%U 74627,105017,147776,207949,292636,411813,579515,815499,1147585,1614917,2272566,3198016,4500318,6332952,8911902,12541080
%N Expansion of 1/(1 - Sum_{k>=0} x^(3*k*(k+1)/2+1)).
%C Number of compositions (ordered partitions) into сentered triangular numbers (A005448).
%C Conjecture: every number > 1 is the sum of at most 5 сentered triangular numbers.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: 1/(1 - Sum_{k>=0} x^(3*k*(k+1)/2+1)).
%F a(n) ~ c / r^n, where r = 0.71061790420456638132596657780064392952867377958... is the root of the equation r^(5/8)*EllipticTheta(2, 0, r^(3/2)) = 2 and c = 0.478786567198436133936216342628844283927491282611910379922933700360643... . - _Vaclav Kotesovec_, Feb 17 2017
%e a(7) = 5 because we have [4, 1, 1, 1], [1, 4, 1, 1], [1, 1, 4, 1], [1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1].
%t nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(3 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]
%Y Cf. A005448, A023361, A280950.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Feb 16 2017
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