A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].

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

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)

Michael Somos, Introduction to Ramanujan theta functions

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 * A348648 A037865 A039969

Adjacent sequences: A028950 A028951 A028952 * A028954 A028955 A028956

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved