0,2

Euler transform of period 12 sequence [2,-3,4,-1,2,-6,2,-1,4,-3,2,-2,...].

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.

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)theta_3(q^3).

The number of nonnegative solutions is given by A119395. - Max Alekseyev (maxale(AT)gmail.com), May 16 2006

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.

J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 110.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).

M. D. Hirschhorn, The number of representations of a number by various forms, Discr. Math., 298 (2005), 205-211.

Michael Gilleland, Some Self-Similar Integer Sequences

M. D. Hirschhorn, Three classical results on representations of a number

Fine gives an explicit formula for a(n) in terms of the divisors of n.

Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+3*j^2).

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]

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

G.f.: theta_3(q)theta_3(q^3) = (Sum_{k} x^(k^2))(Sum_{k} x^(3k^2)).

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

(PARI) a(n)=if(n<1, n==0, qfrep([1, 0; 0, 3], n)[n]*2) /* Michael Somos Jun 05 2005 */

(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 */

(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

Cf. A096936(n)=a(n)/2, if n>0.

Sequence in context: A024308 A059432 A113772 this_sequence A115978 A033751 A033745

Adjacent sequences: A033713 A033714 A033715 this_sequence A033717 A033718 A033719

nonn

N. J. A. Sloane (njas(AT)research.att.com).