A119395 Number of nonnegative integer solutions to the equation x^2 + 3y^2 = n.

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

Offset: 0

Author

Max Alekseyev, May 16 2006

Keywords

• nonn

Comments

The number of integer solutions is given by A033716.

Records 1, 2, 3, 5, 6, 9, 12, 14, 18, ... occur at 0, 4, 28, 196, 364, 2548, 6916, 33124, 48412, ... - Antti Karttunen, Nov 20 2017

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A033716, A096936.

Programs

Mathematica

QP = QPochhammer;
s = (QP[q^2]*QP[q^6])^5/(QP[q]*QP[q^3]*QP[q^4]*QP[q^12])^2 + O[q]^105;
A033716 = CoefficientList[s, q];
A119395 = Ceiling[A033716/4] (* Jean-François Alcover, Jul 02 2018 *)

PARI

{ A033716(n) = local(f,B); f=factorint(n); B=1; for(i=1,matsize(f)[1], if(f[i,1]%3==1,B*=f[i,2]+1); if(f[i,1]%3==2,if(f[i,2]%2,return(0)))); if(n%4,2*B,6*B) } { a(n) = ceil(A033716(n)/4) }

Formula

For n > 0, a(n) = (A033716(n) + 2)/4 if n is a square or a triple of a square; otherwise a(n) = A033716(n)/4. Alternatively, a(n) = ceiling(A033716(n)/4).

G.f.: (1 + theta_3(q))*(1 + theta_3(q^3))/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018