A294624 Number of partitions of n into distinct generalized octagonal numbers (A001082).

1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 2, 3, 1, 1, 2, 2, 1, 0, 1, 3, 2, 2, 3, 1, 1, 1, 3, 5, 2, 2, 3, 2, 2, 1, 3, 5, 2, 1, 3, 3, 2, 1, 3, 6, 3, 1, 3, 4, 3, 1, 4, 7, 3, 0, 3, 6, 4, 1, 2, 7, 5, 2, 4, 5, 5, 2

Offset

0,22

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Vaclav Kotesovec, Graph - the asymptotic ratio (50000 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)))*(1 + x^(k*(3*k+2))).

a(n) ~ ((sqrt(2) - 1)*zeta(3/2))^(1/3) * exp(Pi^(1/3) * (3*(sqrt(2) - 1)*zeta(3/2))^(2/3) * n^(1/3)/2) / (2^(3/2) * 3^(2/3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Mar 11 2026

Example

a(21) = 2 because we have [21] and [16, 5].

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(k (3 k - 2))) (1 + x^(k (3 k + 2))), {k, 1, Floor[Sqrt[1 + 3*nmax]/3] + 1}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 11 2026 *)

Crossrefs

Cf. A001082, A024940, A279281, A290942, A294622, A294623.

Sequence in context: A174294 A089385 A124407 * A290531 A376629 A206442

Adjacent sequences: A294621 A294622 A294623 * A294625 A294626 A294627

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 05 2017

Status

approved