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
KEYWORD
nonn
AUTHOR
STATUS
approved