A222301 Number of points in n-th shell of mcc lattice.

1, 8, 4, 2, 4, 8, 16, 8, 8, 4, 8, 8, 8, 2, 16, 16, 8, 16, 8, 8, 4, 16, 4, 8, 8, 8, 16, 8, 8, 16, 16, 16, 16, 2, 24, 8, 8, 8, 8, 16, 8, 8, 16, 16, 16, 24, 4, 8, 8, 16, 8, 16, 16, 8, 4, 16, 16, 16, 8, 16, 8, 8, 16, 24, 16, 8, 16, 8, 8, 2, 16, 16, 16, 16, 8, 8, 16, 16, 8, 8, 12, 16, 16, 16, 16, 8, 24, 8, 16, 16, 16, 16, 8

Offset

0,2

Comments

The mcc lattice is generated by the vectors (u,v,0), (u,0,v) and (0,v,v), where u = 2^(-1/2), v = 2^(-1/4).

The norms q = X.X of the lattice points X have the form q = s/2 + t/sqrt(2) for integers s and t.

A222301 gives the number of points with each successive value of q; A222302 and A222303 give the corresponding values of s and t.

The theta series of the mcc lattice can then be written as

Sum_{n >= 0} a(n)*z^(s(n)/2+t(n)/sqrt(2)).

Links

Table of n, a(n) for n=0..92.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd. ed., 1993. p. xxiv. (Note that the second set of generators should be [0, +-v, +-v].)

J. H. Conway and N. J. A. Sloane, On lattices equivalent to their duals, J. Number Theory 48 (1994) 373-382.

J. H. Conway and N. J. A. Sloane, The Optimal Isodual Lattice Quantizer in Three Dimensions, Advances in Math. of Commun., Vol. 1, No. 2 (2007), 257-260; arXiv:math/0701080 [math.NT], 2007.

G. Nebe and N. J. A. Sloane, Home page for mcc lattice.

Warren D. Smith, The theta series of the (det=1, isodual) MCC lattice. [Gives first 775 terms.]

Crossrefs

Cf. A222302, A222303.

Sequence in context: A097529 A114321 A154434 * A198353 A010523 A231534

Adjacent sequences: A222298 A222299 A222300 * A222302 A222303 A222304

Keyword

nonn

Author

N. J. A. Sloane, Feb 14 2013

Extensions

a(18) onwards computed by Warren D. Smith.

Status

approved