A333815 G.f.: Sum_{k>=1} x^(k*(3*k - 1)/2) / (1 - x^(3*k)).

1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1

Offset: 1

Author

Ilya Gutkovskiy, Apr 06 2020

Keywords

• nonn

Comments

Number of ways to write n as the difference of two pentagonal numbers.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000326, A001227, A034178, A255849, A333816, A333817, A333818.

Programs

Mathematica

nmax = 92; CoefficientList[Series[Sum[x^(k (3 k - 1)/2)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

A333815list(up_to_n) = { my(s=Ser(sum(k=1,up_to_n,'x^(k*(3*k - 1)/2) / (1 - 'x^(3*k))), 'x, 1+up_to_n)); vector(up_to_n,i,polcoeff(s,i)); }; \\ - Antti Karttunen, Jan 17 2025

PARI

A000326(n) = (n*(3*n-1)/2);
A333815(n) = { my(u=1+floor((n-1)/3), s=0); forstep(i=u,0,-1,my(p2=A000326(i)); if(p2Antti Karttunen, Jan 17 2025

Formula

G.f.: Sum_{i>=0} Sum_{j>=i} Product_{k=i..j} x^(3*k + 1).

Extensions

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025