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

1,22

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

Formula

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

Mathematica

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

Prog

(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(p2<n, return(s)); s += ispolygonal(p2-n, 5)); (s); }; \\ Antti Karttunen, Jan 17 2025

Crossrefs

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

Sequence in context: A325699 A293513 A015738 * A308103 A368987 A115604

Adjacent sequences: A333812 A333813 A333814 * A333816 A333817 A333818

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 06 2020

Extensions

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

Status

approved