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%I #4 Aug 03 2018 08:18:33
%S 1,0,0,2,2,0,0,4,0,0,0,0,2,0,0,0,6,0,0,4,0,0,0,0,0,0,0,2,4,0,0,4,0,0,
%T 0,0,2,0,0,4,0,0,0,4,0,0,0,0,6,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,6,0,0,4,
%U 0,0,0,0,0,0,0,2,4,0,0,4,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,2
%N Expansion of theta_3(q^3)*theta_3(q^4), where theta_3() is the Jacobi theta function.
%C Number of integer solutions to the equation 3*x^2 + 4*y^2 = n.
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F G.f.: Product_{k>=1} (1 + x^(6*k-3))^2*(1 - x^(6*k))*(1 + x^(8*k-4))^2*(1 - x^(8*k)).
%e G.f. = 1 + 2*q^3 + 2*q^4 + 4*q^7 + 2*q^12 + 6*q^16 + 4*q^19 + 2*q^27 + 4*q^28 + ...
%t nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q^3] EllipticTheta[3, 0, q^4], {q, 0, nmax}], q]
%t nmax = 100; CoefficientList[Series[QPochhammer[-q^3, -q^3] QPochhammer[-q^4, -q^4]/(QPochhammer[q^3, -q^3] QPochhammer[q^4, -q^4]), {q, 0, nmax}], q]
%Y Cf. A000049, A020677, A068229, A108563, A192323.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Aug 02 2018
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