cp's OEIS Frontend

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

%I A119395 #17 Aug 01 2018 09:43:56
%S A119395 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,
%T A119395 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,
%U A119395 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
%N A119395 Number of nonnegative integer solutions to the equation x^2 + 3y^2 = n.
%C A119395 The number of integer solutions is given by A033716.
%C A119395 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
%H A119395 Antti Karttunen, <a href="/A119395/b119395.txt">Table of n, a(n) for n = 0..65537</a>
%H A119395 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F A119395 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).
%F A119395 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
%t A119395 QP = QPochhammer;
%t A119395 s = (QP[q^2]*QP[q^6])^5/(QP[q]*QP[q^3]*QP[q^4]*QP[q^12])^2 + O[q]^105;
%t A119395 A033716 = CoefficientList[s, q];
%t A119395 A119395 = Ceiling[A033716/4] (* _Jean-François Alcover_, Jul 02 2018 *)
%o A119395 (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) }
%Y A119395 Cf. A033716, A096936.
%K A119395 nonn
%O A119395 0,5
%A A119395 _Max Alekseyev_, May 16 2006