A014458 Theta series of quadratic form with Gram matrix [ 2, 1, 0; 1, 4, 1; 0, 1, 2 ].

1, 4, 12, 0, 20, 8, 10, 8, 28, 4, 32, 16, 0, 8, 40, 8, 52, 8, 12, 16, 40, 0, 40, 16, 26, 20, 64, 0, 40, 24, 0, 24, 60, 8, 72, 16, 20, 24, 40, 0, 64, 24, 32, 16, 80, 8, 80, 16, 0, 28, 84, 16, 40, 24, 10, 16, 104, 0, 96, 32, 40, 24, 40, 8, 116, 32, 0, 32, 40, 16, 80, 48, 28, 16, 128, 0, 80

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is the tetragonal I lattice (the even holotype) of dimension 3.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 + 4*x + 12*x^2 + 20*x^4 + 8*x^5 + 10*x^6 + 8*x^7 + 28*x^8 + 4*x^9 + ...
1 + 4*q^2 + 12*q^4 + 20*q^8 + 8*q^10 + 10*q^12 + 8*q^14 + 28*q^16 + 4*q^18 + ...
		

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A034933.

Programs

Mathematica

terms = 77; s = Normal[EllipticTheta[3, 0, q]^2*EllipticTheta[3, 0, q^3] + O[q]^(3*terms)][[1 ;; 2 terms]]; Partition[CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017 *)

PARI

{a(n) = if( n<1, n==0, qfrep( [ 2, 1, 0; 1, 4, 1; 0, 1, 2], n, 1)[n] * 2 )} /* Michael Somos, Apr 05 2012 */

PARI

{a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 1, 0; 0, 0, 3], n, 1)[n] * 2 )} /* Michael Somos, Apr 05 2012 */

Formula

a(n) = coefficient of q^2n in theta3(q)^2*theta3(q^3). Theta series of even sublattice of Z^2+sqrt(3)Z - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002

Expansion of (phi(q)^2 * phi(q^3) + phi(-q)^2 * phi(-q^3)) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Apr 05 2012

a(n) = A034933(2*n). - Michael Somos, Apr 05 2012