A008657 Theta series of direct sum of 6 copies of hexagonal lattice.

1, 36, 540, 4356, 20556, 60696, 137916, 325152, 658476, 1023012, 1999080, 3112560, 4446828, 7207992, 10755936, 13150296, 20963052, 27538056, 33706908, 47989008, 64050696, 70696224, 103079952, 124752096, 142308684, 189312156, 237450312, 248276484, 344385504, 397677816

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Crossrefs

Cf. A004016.

Programs

Mathematica

terms = 23; s = ((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)/(3*EllipticTheta[3, 0, q^3]))^6 + O[q]^(2 terms); CoefficientList[s, q^2] (* Jean-François Alcover, Jul 08 2017, from LatticeData(A2) *)

Extensions

More terms from Seiichi Manyama, Oct 19 2018