A035019 Sizes of successive shells in hexagonal (or A_2) lattice.

1, 6, 6, 6, 12, 6, 6, 12, 6, 12, 12, 6, 6, 12, 12, 6, 12, 12, 12, 6, 18, 12, 12, 12, 12, 6, 12, 12, 6, 12, 12, 6, 12, 24, 12, 12, 6, 12, 6, 12, 12, 12, 12, 6, 12, 12, 12, 24, 12, 6, 18, 12, 12, 12, 12, 12, 18, 12, 12, 12, 12, 12, 12, 6, 12, 18, 12, 12, 12, 12

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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. 111.

Links

• T. D. Noe, 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. A003136, A004016, A038590 (partial sums), A357112.

Programs

Maple

S:=series(JacobiTheta2(0,q)*JacobiTheta2(0,q^3)+JacobiTheta3(0,q)*JacobiTheta3(0,q^3),q,1001):
subs(0=NULL,[seq(coeff(S,q,j),j=0..1000)]); # Robert Israel, Jul 29 2016

Mathematica

s = EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^3] + EllipticTheta[3, 0, q]* EllipticTheta[3, 0, q^3] + O[q]^1000; CoefficientList[s, q] /. 0 -> Nothing (* Jean-François Alcover, Sep 19 2016, after Robert Israel *)

Formula

Nonzero coefficients in expansion of theta_3(q)*theta_3(q^3) + theta_2(q)*theta_2(q^3).

The corresponding powers of q are A003136. - Robert Israel, Jul 29 2016