0,2

Theta series of lattice C2 with Gram matrix [1,0; 0,2].

Euler transform of period 8 sequence [2,-1,2,-4,2,-1,2,-2,...].

Expansion of (eta(q^2)eta(q^4))^3/(eta(q)eta(q^8))^2 in powers of q.

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. 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. (32.24).

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.

J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.

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

John Cannon, Table of n, a(n) for n = 0..10000

Michael Gilleland, Some Self-Similar Integer Sequences

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+2*j^2).

G.f. = s(2)^3*s(4)^3/(s(1)^2*s(8)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

G.f.: 1+2 Sum_{k>0} kronecker(-8, n)x^k/(1-x^k) = 1+2 Sum_{k>0} (x^k+x^(3k))/(1+x^(4k)).

G.f.: theta_3(q)theta_3(q^2) = Product_{k>0} (1+x^(2k))((1+x^k)(1-x^(2k))/(1+x^(4k)))^2.

Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...]. - Michael Somos Oct 23 2006

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u1-3*u3)*(u1-u2-u3+u6) -(u2-3*u6)*(u1-2*u2-u3+2*u6) . - Michael Somos Oct 23 2006

d:=proc(r, m, n) local i, t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1, 8, n)+d(3, 8, n)-d(5, 8, n)-d(7, 8, n)), n=1..120)];

(PARI) a(n)=if(n<=0, n==0, 2*(issquare(n)-issquare(2*n)+2*sum(i=1, sqrtint(n\2), issquare(n-2*i^2))))

(PARI) a(n)=if(n<1, n==0, 2*sumdiv(n, d, kronecker(-8, d))) /* Michael Somos Aug 23 2005 */

(PARI) a(n)=if(n<1, n==0, 2*qfrep([1, 0; 0, 2], n)[n]) /* Michael Somos Aug 23 2005 */

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x+A)^-2*eta(x^2+A)^3*eta(x^4+A)^3*eta(x^8+A)^-2, n))}

a(n)=2*A002325(n) for n>0.

Sequence in context: A114427 A129355 A080963 this_sequence A082564 A133692 A139093

Adjacent sequences: A033712 A033713 A033714 this_sequence A033716 A033717 A033718

nonn

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