%I #34 Sep 08 2022 08:44:50
%S 1,0,2,2,2,0,2,0,2,2,0,0,4,2,0,0,4,0,4,0,0,0,0,0,6,0,2,2,0,2,0,2,4,0,
%T 0,0,6,0,0,2,0,2,0,0,0,0,2,2,6,0,2,0,4,0,6,0,0,0,2,0,0,0,2,0,4,0,0,0,
%U 0,2,0,2,8,2,0,2,0,0,6,0,0,4,2,0,0,0,0,2,0,0,0,0,2,2,2,0,8,0,2,0,2,0,0,0,6
%N Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 6 ].
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C The number of integer solutions to n = 2x^2 + xy + 3y^2. - _Michael Somos_, Oct 18 2005
%C In Osburn and Sahu (2010) the g.f. A(q) is denoted by F(z) where q = exp(2 pi i z). - _Michael Somos_, Sep 25 2013
%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 Phi_1, p. 195.
%H John Cannon, <a href="/A028930/b028930.txt">Table of n, a(n) for n = 0..5000</a>
%H Robert Osburn, Brundaban Sahu, <a href="http://arxiv.org/abs/0912.0173">Congruences via modular forms</a>, arXiv:0912.0173 [math.NT], (Sep 02 2010)
%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 G.f.: Sum_{i,j in Z} x^(2*i*i + i*j + 3*j*j). (This is the definition.) - _Michael Somos_, Sep 25 2013
%F Expansion of phi(q^2) * phi(q^46) + 2*q^3 * psi(q) * psi(q^23) + 4*q^12 * psi(q^4) * psi(q^92) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Sep 25 2013, corrected by _Sean A. Irvine_, Feb 13 2020
%F G.f. A(q) = f(t_2(q)) where f() is the g.f. for A224530 and t_2(q) = eta(q) * eta(q^23) / A(q). - _Michael Somos_, Sep 25 2013
%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_, Sep 25 2013
%e For n=24 the solutions are [2,2], [3,-2], [3,1] and their negatives, so a(24)=6.
%e G.f. = 1 + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^6 + 2*x^8 + 2*x^9 + 4*x^12 + ...
%e G.f. = 1 + 2*q^4 + 2*q^6 + 2*q^8 + 2*q^12 + 2*q^16 + 2*q^18 + 4*q^24 + 2*q^26 + 4*q^32 + 4*q^36 + 6*q^48 + 2*q^52 + 2*q^54 + 2*q^58 + 2*q^62 + 4*q^64 + 6*q^72 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^46] + EllipticTheta[ 2, 0, q^2] EllipticTheta[ 2, 0, q^46] + (1/2) EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(23/2)], {q, 0, n}]; (* _Michael Somos_, Sep 25 2013 *)
%t terms = 105; max = Sqrt[terms] // Ceiling; s = Sum[x^(2 i^2 + i*j + 3 j^2), {i, -max, max}, {j, -max, max}]; CoefficientList[s, x][[1 ;; terms]] (* _Jean-François Alcover_, Jul 07 2017, after _Michael Somos_ *)
%o (PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [4, 1; 1, 6], n, 1)[n])}; /* _Michael Somos_, Oct 18 2005 */
%o (PARI) list(n)=concat(1,2*Vec(qfrep([4,1;1,6],n,1))) \\ _Charles R Greathouse IV_, Sep 25 2013
%o (Magma) A := Basis( ModularForms( Gamma1(23), 1), 116); A[1] + 2*A[3] +2*A[4] +2*A[5] +2*A[7] + 2*A[9] + 2*A[10]; /* _Michael Somos_, Aug 24 2014 */
%Y Cf. A030199, A106867, A224530.
%K nonn
%O 0,3
%A _N. J. A. Sloane_.