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%I #15 Mar 12 2021 22:24:41
%S 1,2,0,0,2,0,4,0,4,2,0,0,4,0,0,0,2,0,4,0,0,0,0,2,4,2,4,4,0,0,0,0,4,0,
%T 0,0,6,0,0,4,0,0,0,0,0,0,0,0,8,2,0,0,4,0,4,0,0,0,4,4,0,0,4,0,6,0,0,0,
%U 0,0,0,0,8,0,0,0,0,0,4,0,0,2,4,0,0,0,0,4,0,0,0,0,2,4,4,0,8,0,0
%N Theta series of quadratic form with Gram matrix [ 2, 1; 1, 12 ].
%C theta[2,1;1,2d](z)=theta_3(z)*theta_3((4d-1)z)+theta_2(z)*theta_2((4d-1)z), generalizing the formula for theta(A_2), which is the case d=1 - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 16 2000.
%C The number of integer solutions (x, y) to x^2 + x*y + 6*y^2 = n, discriminant -23. - _Ray Chandler_, Jul 12 2014
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See F_1, p. 195.
%H John Cannon, <a href="/A028959/b028959.txt">Table of n, a(n) for n = 0..5000</a>
%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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of phi(x) * phi(x^23) + 4*x^6 * psi(x^2) * psi(x^46) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Mar 28 2015
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Mar 28 2015
%F G.f.: (theta_3(z)*theta_3(23z) + theta_2(z)*theta_2(23z)).
%e G.f. = 1 + 2*x + 2*x^4 + 4*x^6 + 4*x^8 + 2*x^9 + 4*x^12 + 2*x^16 + 4*x^18 + ...
%e G.f. = 1 + 2*q^2 + 2*q^8 + 4*q^12 + 4*q^16 + 2*q^18 + 4*q^24 + 2*q^32 + 4*q^36 + 2*q^46 + 4*q^48 + 2*q^50 + 4*q^52 + 4*q^54 + 4*q^64 + 6*q^72 + 4*q^78 + 8*q^96 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^23] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^23], {x, 0, n}]; (* _Michael Somos_, Mar 28 2015 *)
%Y Cf. A028958.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
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