%I A033716
%S A033716 1,2,0,2,6,0,0,4,0,2,0,0,6,4,0,0,6,0,0,4,0,4,0,0,0,2,0,2,12,0,0,4,0,0,
               0,
%T A033716 0,6,4,0,4,0,0,0,4,0,0,0,0,6,6,0,0,12,0,0,0,0,4,0,0,0,4,0,4,6,0,0,4,0,
               0,
%U A033716 0,0,0,4,0,2,12,0,0,4,0,2,0,0,12,0,0,0,0,0,0,8,0,4,0,0,0,4,0,0,6,0
%N A033716 Number of integer solutions to the equation x^2+3y^2=n.
%C A033716 Euler transform of period 12 sequence [2,-3,4,-1,2,-6,2,-1,4,-3,2,-2,
               ...].
%C A033716 Expansion of (eta(q^2)eta(q^6))^5/(eta(q)eta(q^3)eta(q^4)eta(q^12))^2 
               in powers of q.
%C A033716 The cubic modular equation for k is equivalent to theta_4(q)theta_4(q^3)+theta_2(q)theta_2(q^3)=theta_3(q)the\
               ta_3(q^3).
%C A033716 The number of nonnegative solutions is given by A119395. - Max Alekseyev 
               (maxale(AT)gmail.com), May 16 2006
%D A033716 G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series 
               to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
%D A033716 J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 110.
%D A033716 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p 102 eq 9.
%D A033716 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 78, Eq. (32.25).
%D A033716 M. D. Hirschhorn, The number of representations of a number by various 
               forms, Discr. Math., 298 (2005), 205-211.
%H A033716 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%H A033716 M. D. Hirschhorn, <a href="http://www.mat.univie.ac.at/~slc/opapers/s42hirsch.html">
               Three classical results on representations of a number</a>
%F A033716 Fine gives an explicit formula for a(n) in terms of the divisors of n.
%F A033716 Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+3*j^2).
%F A033716 G.f. = s(2)^5*s(6)^5/(s(1)^2*s(3)^2*s(4)^2*s(12)^2), where s(k) := subs(q=q^k, 
               eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A033716 G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u1, u3, u9)=(u1*u9)*(u1^2-3*u1*u3+3*u3^2)*(u3^2-3*u3*u9\
               +3*u9^2)-u3^6 . - Michael Somos Sep 05 2005
%F A033716 G.f.: theta_3(q)theta_3(q^3) = (Sum_{k} x^(k^2))(Sum_{k} x^(3k^2)).
%F A033716 Let n=3^d*p1^(2*b1)*...*pm^(2*bm)*q1^c1*...*qk^ck be a prime factorization 
               of n where pi are primes of the form 3t+2 and qj are primes of the 
               form 3t+1. Let B=(c1+1)*...*(ck+1). Then a(n)=0 if either of bi is 
               a half-integer; a(n)=6B if n is a multiple of 4; and a(n)=2B otherwise. 
               - Max Alekseyev (maxale(AT)gmail.com), May 16 2006
%o A033716 (PARI) a(n)=if(n<1, n==0, qfrep([1,0;0,3],n)[n]*2) /* Michael Somos Jun 
               05 2005 */
%o A033716 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A))^5/
               (eta(x+A)*eta(x^3+A)*eta(x^4+A)*eta(x^12+A))^2, n))} /* Michael Somos 
               Jun 05 2005 */
%o A033716 (PARI) { a(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) } - Max Alekseyev (maxale(AT)gmail.com), May 16 2006
%Y A033716 Cf. A096936(n)=a(n)/2, if n>0.
%Y A033716 Sequence in context: A024308 A059432 A113772 this_sequence A115978 A033751 
               A033745
%Y A033716 Adjacent sequences: A033713 A033714 A033715 this_sequence A033717 A033718 
               A033719
%K A033716 nonn
%O A033716 0,2
%A A033716 N. J. A. Sloane (njas(AT)research.att.com).