0,4

Number of solutions of 8n+4=x^2+3y^2 in positive odd integers. - Michael Somos Sep 18 2004

Given g.f. A(x), then q^(1/2)*2*A(q) is denoted phi_1(z) where q=exp(pi*i*z) in Conway and Sloane.

Half of theta series of planar hexagonal lattice (A2) with respect to edge.

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223 Entry 3(i).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103. see Equ. (13)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

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

Expansion of q^(-1/2)(eta(q^2)eta(q^6))^2/(eta(q)eta(q^3)) in powers of q. - Michael Somos Apr 18 2004

Euler transform of period 6 sequence [1,-1,2,-1,1,-2,...]. - Michael Somos Apr 18 2004

G.f. = s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

G.f.: (Sum_{j>0} x^((j^2-j)/2))(Sum_{k>0} x^(3(k^2-k)/2)) = Product_{k>0} (1+x^k)(1-x^(2k))(1+x^(3k))(1-x^(6k)). - Michael Somos Sep 18 2004

G.f.: Sum_{k>=0} a(k)x^(2k+1) = Sum_{k>0} x^k/(1+x^k+x^(2k)) -x^(4k)/(1+x^(4k)+x^(8k)) . - Michael Somos Nov 04 2005

Given g.f. A(x), then B(x)=(x*A(x^2))^2 satisfies 0=f(B(x),B(x^2),B(x^4)) where f(u,v,w)=v^3+4uvw+16vw^2-8wv^2-wu^2. - Michael Somos Sep 18 2004

Expansion of q^(-1)*(a(q)-a(q^4))/6 in powers of q^2 where a() is a cubic AGM analog function. - Michael Somos Oct 24 2006

Givn g.f. A(x), then xA(x^2) = Sum_{k>0} x^k(1-x^k)(1-x^(4k))(1-x^(5k))/(1-x^(12k)). - Michael Somos Sep 18 2004

Multiplicative with a(n)=b(2*n+1) and b(2^e)=0^e, b(3^e)=1, b(p^e)=(1+(-1)^e)/2 if p==5 mod 6 otherwise b(p^e)=e+1. - Michael Somos Sep 18 2004. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x*A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)

a(6)=2 since 8*6+4=52=5^2+3*3^2=7^2+3*1^2.

(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff((eta(x^2+A)*eta(x^6+A))^2/(eta(x+A)*eta(x^3+A)), n)) /* Michael Somos Sep 18 2004 */

(PARI) a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-12, d)*(n/d%2))) /* Michael Somos Nov 04 2005 */

(PARI) a(n)=if(n<0, 0, n=8*n+4; sum(j=1, sqrtint(n\3), (j%2)*issquare(n-3*j^2))) /* Michael Somos Nov 04 2005 */

a(n) = A093829(2n+1) = A035178(2n+1). A005881(n) = 2*a(n).

Sequence in context: A112214 A112608 A058677 this_sequence A129449 A033798 A033792

Adjacent sequences: A033759 A033760 A033761 this_sequence A033763 A033764 A033765

nonn,mult

njas