A279281 Expansion of Product_{k>=1} (1 + x^(k*(3*k-2))).

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1

Offset

0,106

Comments

Number of partitions of n into distinct octagonal numbers (A000567).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Vaclav Kotesovec, Graph - the asymptotic ratio (200000 terms)

Eric Weisstein's World of Mathematics, Octagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

Formula

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

a(n) ~ zeta(3/2)^(1/3) * (sqrt(2) - 1)^(1/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * (sqrt(2) - 1)^(2/3) * n^(1/3) / (2^(5/3)*d^(1/3))) / (2^(4/3 + b/(2*d)) * sqrt(3) * d^(1/6) * Pi^(1/3) * n^(5/6)) * (1 - ((sqrt(2) - 1)^(4/3) * b^2 * Pi^(2/3) * zeta(1/2) * zeta(3/2)^(1/3) / (2^(23/6) * d^(5/3)) + 5*d^(1/3) / (9 * (2*Pi)^(1/3) * (sqrt(2) - 1)^(2/3) * zeta(3/2)^(2/3))) / n^(1/3)), where d = 3, b = -2. - Vaclav Kotesovec, Mar 11 2026

Example

a(105) = 2 because we have [96, 8, 1] and [65, 40].

Mathematica

nmax = 120; CoefficientList[Series[Product[1 + x^(k*(3*k-2)), {k, 1, Sqrt[1 + 3*nmax]/3 + 1}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 10 2026 *)

Crossrefs

Cf. A000567, A024940, A033461, A218380, A279041, A279279, A279280.

Sequence in context: A377336 A181563 A056976 * A342465 A124749 A350890

Adjacent sequences: A279278 A279279 A279280 * A279282 A279283 A279284

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 09 2016

Status

approved