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Theta series of direct sum of 6 copies of hexagonal lattice.
2

%I #17 Oct 19 2018 08:06:08

%S 1,36,540,4356,20556,60696,137916,325152,658476,1023012,1999080,

%T 3112560,4446828,7207992,10755936,13150296,20963052,27538056,33706908,

%U 47989008,64050696,70696224,103079952,124752096,142308684,189312156,237450312,248276484,344385504,397677816

%N Theta series of direct sum of 6 copies of hexagonal lattice.

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

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

%H Seiichi Manyama, <a href="/A008657/b008657.txt">Table of n, a(n) for n = 0..10000</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>

%t 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) *)

%Y Cf. A004016.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Seiichi Manyama_, Oct 19 2018