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%I #18 Aug 12 2019 02:23:13
%S 1,4,4,2,12,16,0,8,20,4,8,8,10,32,8,0,28,24,4,8,32,16,16,16,0,28,8,2,
%T 40,48,8,8,52,0,8,16,12,64,16,8,40,24,0,24,40,16,16,16,26,28,20,0,64,
%U 80,0,16,40,24,24,8,0,64,24,8,60,48,8,24,72,0,16,16,20,48,24,10,40,96
%N Expansion of theta_3(q)^2 * theta_3(q^3) in powers of q.
%C a(n)=0 if n == 6*9^k (mod 9^(k+1)) for some k>=0. - _Robert Israel_, Aug 11 2019
%H Robert Israel, <a href="/A034933/b034933.txt">Table of n, a(n) for n = 0..10000</a>
%F Number of integer solutions to x^2 + y^2 + 3*z^2 = n.
%F Euler transform of period 12 sequence [4, -6, 6, -2, 4, -9, 4, -2, 6, -6, 4, -3, ...]. - _Michael Somos_, Sep 21 2005
%F Expansion of (eta(q^2)^2 * eta(q^6))^5 / (eta(q)^2 * eta(q^3) * eta(q^4)^2 * eta(q^12))^2 in power of q. - _Michael Somos_, Sep 21 2005
%F G.f.: theta_3(q)^2 * theta_3(q^3).
%e 1 + 4*q + 4*q^2 + 2*q^3 + 12*q^4 + 16*q^5 + 8*q^7 + 20*q^8 + 4*q^9 +...
%p S:= series(JacobiTheta3(0,q)^2*JacobiTheta3(0,q^3),q,101):
%p seq(coeff(S,q,i),i=0..100); # _Robert Israel_, Aug 11 2019
%t CoefficientList[EllipticTheta[3, 0, q]^2*EllipticTheta[3, 0, q^3]+O[q]^80, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 1, 0; 0, 0, 3], n)[n] * 2)} /* _Michael Somos_, Sep 21 2005 */
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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