A014452 Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 2, 1; 0, 1, 2 ].

1, 2, 6, 12, 2, 0, 18, 12, 6, 14, 12, 12, 12, 0, 12, 36, 2, 12, 42, 12, 0, 0, 24, 24, 18, 14, 12, 48, 12, 0, 48, 12, 6, 36, 12, 24, 14, 0, 24, 48, 12, 12, 72, 36, 12, 0, 24, 24, 12, 14, 30, 72, 0, 0, 54, 24, 12, 48, 36, 12, 36, 0, 36, 84, 2, 24, 48, 36, 12, 0, 24, 24, 42, 24, 36, 60, 12

Offset

0,2

Comments

This is the hexagonal P lattice (the classical holotype) of dimension 3.

Links

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

G. Nebe and N. J. A. Sloane, Home page for this lattice

Formula

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 8/3 * Pi^(7/4) * 2^(3/4) / Gamma(2/3)^2 / Gamma(11/12)^(1/2) / Gamma(7/12)^(5/2) / (3^(1/2)-1) / (sqrt(2) * (1+3^(1/2)))^(3/2) = A388238. - Simon Plouffe, Sep 15 2025

Example

1 + 2*q + 6*q^2 + 12*q^3 + 2*q^4 + 18*q^6 + 12*q^7 + 6*q^8 + 14*q^9 + 12*q^10 + ...

Crossrefs

Sequence in context: A293589 A293117 A293122 * A188894 A386922 A355416

Adjacent sequences: A014449 A014450 A014451 * A014453 A014454 A014455

Keyword

nonn

Author

N. J. A. Sloane

Status

approved