A028930 Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 6 ].

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, 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, 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

Offset

0,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The number of integer solutions to n = 2x^2 + xy + 3y^2. - Michael Somos, Oct 18 2005

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

References

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.

Links

John Cannon, Table of n, a(n) for n = 0..5000

Robert Osburn, Brundaban Sahu, Congruences via modular forms, arXiv:0912.0173 [math.NT], (Sep 02 2010)

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

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

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

Example

For n=24 the solutions are [2,2], [3,-2], [3,1] and their negatives, so a(24)=6.
G.f. = 1 + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^6 + 2*x^8 + 2*x^9 + 4*x^12 + ...
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 + ...

Mathematica

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 *)
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 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [4, 1; 1, 6], n, 1)[n])}; /* Michael Somos, Oct 18 2005 */
(PARI) list(n)=concat(1, 2*Vec(qfrep([4, 1; 1, 6], n, 1))) \\ Charles R Greathouse IV, Sep 25 2013
(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 */

Crossrefs

Cf. A030199, A106867, A224530.

Sequence in context: A133625 A176154 A274718 * A112792 A138319 A217864

Adjacent sequences: A028927 A028928 A028929 * A028931 A028932 A028933

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved