A282502 Expansion of 1/(1 - Sum_{k>=0} x^(3*k*(k+1)/2+1)).

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 15, 21, 29, 40, 57, 81, 114, 159, 223, 315, 445, 626, 879, 1236, 1741, 2452, 3450, 4852, 6826, 9608, 13524, 19032, 26778, 37680, 53027, 74627, 105017, 147776, 207949, 292636, 411813, 579515, 815499, 1147585, 1614917, 2272566, 3198016, 4500318, 6332952, 8911902, 12541080

Offset: 0

Author

Ilya Gutkovskiy, Feb 16 2017

Keywords

• nonn

Comments

Number of compositions (ordered partitions) into centered triangular numbers (A005448).

Conjecture: every number > 1 is the sum of at most 5 centered triangular numbers.

Examples

			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].
		

Links

• Eric Weisstein's World of Mathematics, Centered Triangular Number
• Index entries for sequences related to centered polygonal numbers
• Index entries for sequences related to compositions

Crossrefs

Cf. A005448, A023361, A280950.

Programs

Mathematica

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(3 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=0} x^(3*k*(k+1)/2+1)).

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