A294622 Number of partitions of n into generalized octagonal numbers (A001082).

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 8, 8, 9, 9, 10, 13, 13, 14, 16, 17, 20, 20, 21, 24, 25, 28, 31, 33, 36, 37, 40, 45, 47, 50, 55, 59, 65, 67, 70, 77, 81, 87, 94, 99, 107, 111, 117, 127, 133, 141, 152, 160, 172, 178, 186, 201, 210, 223, 237, 249, 267, 276, 289, 308, 322, 341, 360, 378, 401

Offset

0,6

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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/((1 - x^(k*(3*k-2)))*(1 - x^(k*(3*k+2)))).

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

Example

a(8) = 3 because we have [8], [5, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Mathematica

nmax = 74; CoefficientList[Series[Product[1/((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, A007294, A095699, A279041, A294621, A294624.

Sequence in context: A247366 A285760 A025778 * A078451 A332247 A341168

Adjacent sequences: A294619 A294620 A294621 * A294623 A294624 A294625

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 05 2017

Status

approved