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%I
%S 1,2,0,4,2,4,0,0,0,6,0,2,4,0,0,8,2,0,0,0,4,0,0,4,0,6,0,8,0,0,0,4,0,4,
%T 0,0,6,4,0,0,0,0,0,0,2,12,0,4,4,2,0,0,0,4,0,4,0,0,0,4,8,0,0,0,2,0,0,4,
%U 0,8,0,4,0,0,0,12,0,0,0,0,4,10,0,0,0,0,0,0,0,4,0,0,4,8,0,0,0,4,0,6,6,0,0
%N Expansion of (theta_3(z)*theta_3(11z) + theta_2(z)*theta_2(11z)).
%C Theta series of lattice with Gram matrix [2, 1; 1, 6].
%C Number of integer solutions (x, y) to x^2 + x*y + 3*y^2 = n. - Michael Somos, Sep 20 2004
%D H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032)
%H John Cannon, <a href="/A028609/b028609.txt">Table of n, a(n) for n = 0..5000</a>
%F Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...]. - Michael Somos, Jan 29 2007
%F a(n) = 2 * b(n) and b(n) is multiplicative with b(11^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), b(p^e) = e+1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007
%F G.f.: 1 + 2 * Sum_{k>0} kronecker( -11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007
%F G.f. is Fourier series of a weight 1 level 11 modular form. f(-1/ (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos, Jun 05 2007
%F Expansion of (F(x)^2 + 4 * F(x^2)^2 + 8 * F(x^4)^2) / F(x^2) in powers of x or expansion of (F(x)^2 + 2 * F(x^2)^2 + 2 * F(x^4)^2) / F(x^2) in powers of x^4 where F(x) = x^(1/2) * f(-x) * f(-x^11) and f() is a Ramanujan theta function. - Michael Somos, Mar 01 2010
%F a(n) = 2 * A035179(n) unless n=0. Convolution square is A028610.
%e 1 + 2*x + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^9 + 2*x^11 + 4*x^12 + 8*x^15 + ...
%e Theta series of lattice with Gram matrix [2, 1; 1, 6] = 1 + 2*q^2 + 4*q^6 + 2*q^8 + 4*q^10 + 6*q^18 + 2*q^22 + 4*q^24 + 8*q^30 + 2*q^32 + 4*q^40 + 4*q^46 + 6*q^50 + 8*q^54 + 4*q^62 + 4*q^66 + 6*q^72 + 4*q^74 + ...
%o (PARI) {a(n) = local(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n * 4\11), 2 * issquare( t=4*n - 11*y^2) - (t==0)))} /* Michael Somos, Sep 20 2004 */
%o (PARI) {a(n) = if( n<1, n==0, qfrep( [ 2, 1; 1, 6], n, 1)[n] * 2)} /* Michael Somos, Jun 05 2005 */
%o (PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -11, p) * X))[n] * 2)} /* Michael Somos, Jun 05 2005 */
%o (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -11, d)))} /* Michael Somos, Jan 29 2007 */
%Y Cf. A028610, A035179.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
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