A008655 Theta series of direct sum of 4 copies of hexagonal lattice.

1, 24, 216, 888, 1752, 3024, 7992, 8256, 14040, 24216, 27216, 31968, 64824, 52752, 74304, 111888, 112344, 117936, 217944, 164640, 220752, 305472, 287712, 292032, 519480, 378024, 474768, 654072

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.

Convolution of A008654 and A004016. Convolution square of A008653. - R. J. Mathar, Feb 22 2021

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 (terms 0..1000 from G. C. Greubel)
• J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Programs

Maple

A008655 := proc(n)
        add( A004016(i)*x^i,i=0..n) ;
        coeftayl(%^4,x=0,n) ;
end proc: # R. J. Mathar, Feb 22 2021

Mathematica

terms = 28; s = (EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(81*EllipticTheta[3, 0, q^3]^4) + O[q]^(2 terms); CoefficientList[s, q^2][[1 ;; terms]] (* Jean-François Alcover, Jul 07 2017, from LatticeData(A2) *)

Formula

Expansion of (theta_3(z)*theta_3(3z) + theta_2(z)*theta_2(3z))^4.