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 A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ]. 1
 1, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 4, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 8, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 2, 6, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The number of integer solutions (x, y) to 3*x^2 + 2*x*y + 4*y^2, discriminant -44. - Ray Chandler, Jul 12 2014 LINKS John Cannon, Table of n, a(n) for n = 0..10000 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 Expansion of phi(q) * phi(q^11) - 2*q * f(-q^2) * f(-q^22) = phi(q^3) * phi(q^33) + 2*q^4 * chi(q) * psi(-q^3) * chi(q^11) * psi(-q^33) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos and Alex Berkovich, Jun 24 2011 G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 24 2011 G.f.: Sum_{n, m in Z} x ^ (3*n*n + 2*n*m + 4*m*m). a(4*n + 2) = a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0. - Michael Somos, Feb 23 2012 EXAMPLE G.f. = 1 + 2*q^3 + 2*q^4 + 2*q^5 + 2*q^9 + 4*q^12 + 2*q^15 + 2*q^16 + 4*q^20 + 2*q^23 + 2*q^25 + 2*q^27 + 2*q^31 + 2*q^33 + 6*q^36 + 2*q^37 + 2*q^44 + 4*q^45 + ... MATHEMATICA r[n_] := Reduce[{x, y}.{{3, 1}, {1, 4}}.{x, y} == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] - 2 q QPochhammer[ q^2] QPochhammer[ q^22], {q, 0, n}]; (* Michael Somos, Feb 09 2017 *) PROG (PARI) {a(n) = if( n<1, n==0, qfrep([3, 1; 1, 4], n)[n] * 2)}; /* Michael Somos, Jun 24 2011 */ (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\11), 2 * x^(11*k^2), 1 + A) - 2 * x * eta(x^2 + A) * eta(x^22 + A), n))}; /* Michael Somos, Jun 24 2011 */ (MAGMA) A := Basis( ModularForms( Gamma1(44), 1), 87); A[1] + 2*A[4] + 2*A[5] + 2*A[6] + 2*A[10] + 4*A[13] + 2*A[16] + 2*A[17] + 4*A[21] + 2*A[24]; /* Michael Somos, Feb 09 2017 */ CROSSREFS Cf. A028952, A106282. Sequence in context: A214458 A133873 A163326 * A037865 A039969 A039967 Adjacent sequences:  A028950 A028951 A028952 * A028954 A028955 A028956 KEYWORD nonn AUTHOR STATUS approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)