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 A028930 Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 6 ]. 13
 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 (list; graph; refs; listen; history; text; internal format)
 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) 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 + 2*A +2*A +2*A +2*A + 2*A + 2*A; /* 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 STATUS approved

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Last modified August 12 03:34 EDT 2020. Contains 336436 sequences. (Running on oeis4.)