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A222301 Number of points in n-th shell of mcc lattice. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)