A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2*k*(k+1)+1)).

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 21, 28, 37, 49, 65, 88, 119, 160, 214, 285, 381, 511, 687, 923, 1237, 1656, 2217, 2971, 3985, 5345, 7166, 9603, 12867, 17244, 23115, 30989, 41543, 55684, 74634, 100032, 134081, 179729, 240919, 322935, 432858, 580191, 777680, 1042407, 1397262, 1872911, 2510457

Offset: 0

Author

Ilya Gutkovskiy, Feb 16 2017

Keywords

• nonn

Comments

Number of compositions (ordered partitions) into centered square numbers (A001844).

Conjecture: every number > 1 is the sum of at most 6 centered square numbers.

Extended conjecture: every number > 1 is the sum of at most k+2 centered k-gonal numbers.

Examples

			a(8) = 5 because we have [5, 1, 1, 1], [1, 5, 1, 1], [1, 1, 5, 1], [1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1].
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Centered Square Number
• Index entries for sequences related to centered polygonal numbers
• Index entries for sequences related to compositions

Crossrefs

Cf. A001844, A006456, A280951.

Programs

Mathematica

nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(2 k (k + 1) + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

PARI

Vec(1/(1 - sum(k=0, 54, x^(2*k*(k + 1) + 1))) + O(x^54)) \\ Indranil Ghosh, Mar 15 2017

Formula

G.f.: 1/(1 - Sum_{k>=0} x^(2*k*(k+1)+1)).

a(n) ~ c / r^n, where r = 0.746043978237212782246711857485153004976647... is the root of the equation sqrt(r) * EllipticTheta(2, 0, r^2) = 2 and c = 0.453173429667590077751072798128748901015122665... . - Vaclav Kotesovec, Feb 17 2017