A035454 Number of partitions of n into parts 8k+2 or 8k+3.

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 13, 16, 16, 19, 20, 23, 25, 29, 30, 35, 37, 41, 44, 50, 53, 60, 64, 72, 76, 85, 90, 100, 107, 118, 126, 140, 148, 163, 174, 190, 203, 223, 237, 260, 277, 301, 321, 349, 371, 403, 430, 466, 496, 537

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A035685.

Programs

Maple

nmax:= 100:
g:= mul(1/(1-x^(8*k+2))/(1-x^(8*k+3)),k=0..(nmax-2)/8):
S:= series(g,x,nmax+1):
seq(coeff(S,x,j),j=1..nmax); # Robert Israel, Dec 11 2018

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+2))*(1 - x^(8k+3))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 2}~Join~{Range[0, kmax]*8 + 3}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

Formula

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(1/4) * Gamma(3/8) / (8 * 2^(5/16) * 3^(1/16) * Pi^(11/8) * n^(9/16)). - Vaclav Kotesovec, Aug 26 2015

G.f.: Product_{k>=0} 1/((1-x^(8*k+2))*(1-x^(8*k+3))). - Robert Israel, Dec 11 2018