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A008655 Theta series of direct sum of 4 copies of hexagonal lattice. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

FORMULA

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

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) *)

CROSSREFS

Sequence in context: A169635 A269496 A221434 * A133754 A104670 A205968

Adjacent sequences:  A008652 A008653 A008654 * A008656 A008657 A008658

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 7 20:03 EST 2021. Contains 349585 sequences. (Running on oeis4.)