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

Author

N. J. A. Sloane

Keywords

• nonn

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.

Examples

			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 + ...
		

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.

Crossrefs

Cf. A028966.

Programs

Magma

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

Extensions

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