A222301 - OEIS

%I #32 Dec 06 2018 04:37:34

%S 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,

%T 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,

%U 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

%N Number of points in n-th shell of mcc lattice.

%C 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).

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

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

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

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

%H J. H. Conway and N. J. A. Sloane, <a href="https://dx.doi.org/10.1007/978-1-4757-2249-9">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, 3rd. ed., 1993. p. xxiv. (Note that the second set of generators should be [0, +-v, +-v].)

%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1006/jnth.1994.1073">On lattices equivalent to their duals</a>, J. Number Theory 48 (1994) 373-382.

%H J. H. Conway and N. J. A. Sloane, <a href="http://arxiv.org/abs/math/0701080">The Optimal Isodual Lattice Quantizer in Three Dimensions</a>, Advances in Math. of Commun., Vol. 1, No. 2 (2007), 257-260; arXiv:math/0701080 [math.NT], 2007.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/mcc.html">Home page for mcc lattice</a>.

%H Warren D. Smith, <a href="/A222301/a222301.txt">The theta series of the (det=1, isodual) MCC lattice</a>. [Gives first 775 terms.]

%Y Cf. A222302, A222303.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 14 2013

%E a(18) onwards computed by _Warren D. Smith_.