A028967 Theta series of a.c.c. lattice.

1, 0, 10, 4, 0, 8, 12, 0, 26, 0, 0, 8, 20, 0, 32, 8, 0, 16, 10, 0, 40, 8, 0, 16, 28, 0, 40, 4, 0, 8, 32, 0, 58, 16, 0, 16, 0, 0, 72, 8, 0, 16, 40, 0, 40, 8, 0, 32, 52, 0, 50, 8, 0, 24, 12, 0, 64, 16, 0, 24, 40, 0, 96, 0, 0, 16, 40, 0, 80, 16, 0, 16, 26, 0, 40, 20, 0, 32, 64, 0, 104, 0, 0, 40, 40

Offset

0,3

Comments

At one time Conway and I called this the z.c.c. lattice, which has led to some confusion. The a.c.c. and z.c.c. are different names for the same lattice. The a.c.c. name is preferred.

Gram matrix:

[ +4 -1 +1]

[ -1 +4 +2]

[ +1 +2 +4]

This lattice has determinant 36 and kissing number 10; it is not isodual.

Links

John Cannon, Table of n, a(n) for n = 0..5000

J. H. Conway and N. J. A. Sloane, On Lattices Equivalent to Their Duals, J. Number Theory, 48 (1994), 373-382.

K. L. Fields, The fragile lattice packings of spheres in three-dimensional space, Acta Cryst. Sect. A 36 (1980), 194-197.

A. Patterson, Crystal lattice models based on the close packing of spheres, Rev. Sci. Instrumen., 12 (1941), 206-211.

Example

1 + 10*q^4 + 4*q^6 + 8*q^10 + 12*q^12 + 26*q^16 + 8*q^22 + 20*q^24 + 32*q^28 + 8*q^30 + ...

Prog

(Magma) L:=LatticeWithGram(3, [4, -1, -1, -1, 4, -2, -1, -2, 4]); T<q> := ThetaSeries(L, 500); T;

Crossrefs

Cf. A028966.

Sequence in context: A077194 A038305 A089478 * A097530 A063565 A371918

Adjacent sequences: A028964 A028965 A028966 * A028968 A028969 A028970

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

Edited by N. J. A. Sloane, Sep 29 2006

Status

approved