A329265 a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 3.

1, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 4, 0, 0, 4, 2, 8, 0, 4, 18, 0, 0, 8, 0, 4, 0, 4, 12, 0, 0, 0, 4, 2, 0, 8, 4, 0, 0, 0, 0, 8, 0, 4, 16, 0, 0, 12, 4, 4, 0, 0, 16, 0, 0, 8, 10, 16, 0, 8, 16, 0, 0, 0, 4, 18, 0, 0, 16

Offset: 0

Author

Stefano Spezia, Nov 09 2019

Keywords

• nonn

Examples

			a(9) = 6 since there are 6 integer solutions to 1^3*k1^2 + 2^3*k2^2 + ... = 9:
k1 = +-3 and k_j = 0 for j > 1;
k1 = -1, k2 = +-1 and k_j = 0 for j > 2;
k1 = 1, k2 = +-1 and k_j = 0 for j > 2.
		

Links

• Nian Hong Zhou, Yalin Sun, Counting the number of solutions to certain infinite Diophantine equations, arXiv:1910.07884 [math.NT], 2019.

Crossrefs

Cf. A000041, A000122, A320067 (r = 1), A320068, A320078, A320968, A320992, A329264 (r = 2), A329266 (r = 4).

Programs

Mathematica

nmax=85; r=3; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)),{n,1,nmax}],{j,1,nmax}],{q,0,nmax}],q]

Formula

a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 3 (see Proposition 1.1 in Zhou and Sun).