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%I #30 Mar 12 2021 22:24:41
%S 1,4,12,0,20,8,10,8,28,4,32,16,0,8,40,8,52,8,12,16,40,0,40,16,26,20,
%T 64,0,40,24,0,24,60,8,72,16,20,24,40,0,64,24,32,16,80,8,80,16,0,28,84,
%U 16,40,24,10,16,104,0,96,32,40,24,40,8,116,32,0,32,40,16,80,48,28,16,128,0,80
%N Theta series of quadratic form with Gram matrix [ 2, 1, 0; 1, 4, 1; 0, 1, 2 ].
%C This is the tetragonal I lattice (the even holotype) of dimension 3.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H John Cannon, <a href="/A014458/b014458.txt">Table of n, a(n) for n = 0..5000</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/tetragonalIe.html">Home page for this lattice</a>
%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 a(n) = coefficient of q^2n in theta3(q)^2*theta3(q^3). Theta series of even sublattice of Z^2+sqrt(3)Z - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
%F Expansion of (phi(q)^2 * phi(q^3) + phi(-q)^2 * phi(-q^3)) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - _Michael Somos_, Apr 05 2012
%F a(n) = A034933(2*n). - _Michael Somos_, Apr 05 2012
%e 1 + 4*x + 12*x^2 + 20*x^4 + 8*x^5 + 10*x^6 + 8*x^7 + 28*x^8 + 4*x^9 + ...
%e 1 + 4*q^2 + 12*q^4 + 20*q^8 + 8*q^10 + 10*q^12 + 8*q^14 + 28*q^16 + 4*q^18 + ...
%t terms = 77; s = Normal[EllipticTheta[3, 0, q]^2*EllipticTheta[3, 0, q^3] + O[q]^(3*terms)][[1 ;; 2 terms]]; Partition[CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* _Jean-François Alcover_, Jul 04 2017 *)
%o (PARI) {a(n) = if( n<1, n==0, qfrep( [ 2, 1, 0; 1, 4, 1; 0, 1, 2], n, 1)[n] * 2 )} /* _Michael Somos_, Apr 05 2012 */
%o (PARI) {a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 1, 0; 0, 0, 3], n, 1)[n] * 2 )} /* _Michael Somos_, Apr 05 2012 */
%Y Cf. A034933.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
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