A028928 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 5 ].

1, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 6, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 6, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 6, 2, 0, 0, 0, 0, 2, 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 n = 3*x^2 + 2*x*y + 5*y^2, discriminant -56. - Ray Chandler, Jul 12 2014

Links

John Cannon, Table of n, a(n) for n = 0..10000

Michael Gilleland, Some Self-Similar Integer Sequences

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^3) * phi(q^42) + 2*q^5 * chi(q) * psi(-q^3) * chi(q^14) * psi(-q^42) = phi(q^6) * phi(q^21) + 2*q^3 * chi(q^2) * psi(-q^6) * chi(q^7) * psi(-q^21) = phi(q^2) * phi(q^7) - 2*q^2 * phi(-q^4) * psi(q^7) * chi(-q) / chi(-q^28) in powers of q where phi(), psi(), chi() are Ramanujan theta functions - Michael Somos and Alex Berkovich, Jun 06 2011

Expansion of - phi(q) * phi(q^14) + 2 * chi(q) * f(-q^7) * f(-q^8) * chi(q^14) in powers of q where phi(), chi(), f() are Ramanujan theta functions - Michael Somos, Jun 22 2011

G.f.: Sum_{n, m in Z} x^(3*n*n + 2*n*m + 5*m*m).

Example

G.f. = 1 + 2*q^3 + 2*q^5 + 2*q^6 + 2*q^10 + 2*q^12 + 2*q^13 + 2*q^19 + 2*q^20 + 2*q^21 + 2*q^24 + 2*q^26 + 4*q^27 + 2*q^35 + 2*q^38 + 2*q^40 + 2*q^42 + 6*q^45 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], If[ -1 != KroneckerSymbol[ -7, n / 7^IntegerExponent[ n, 7]], 0, Sum[ KroneckerSymbol[ -14, d], { d, Divisors @ n}]]]; (* Michael Somos, Jul 13 2011 *)

Prog

(PARI) {a(n) = if( n<1, n==0, qfrep([3, 1; 1, 5], n)[n] * 2)}; /* Michael Somos, Jun 06 2011 */
(PARI) {a(n) = if( n<1, n==0, (-1 == kronecker( -7, n / 7^valuation( n, 7))) * sumdiv( n, d, kronecker( -14, d)))}; /* Michael Somos, Jun 22 2011 */

Crossrefs

Cf. A106915.

Sequence in context: A174610 A375108 A373924 * A343723 A260149 A091379

Adjacent sequences: A028925 A028926 A028927 * A028929 A028930 A028931

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved