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A028967 Theta series of a.c.c. lattice. 4

%I #16 Sep 08 2022 08:44:50

%S 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,

%T 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,

%U 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

%N Theta series of a.c.c. lattice.

%C 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.

%C Gram matrix:

%C [ +4 -1 +1]

%C [ -1 +4 +2]

%C [ +1 +2 +4]

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

%H John Cannon, <a href="/A028967/b028967.txt">Table of n, a(n) for n = 0..5000</a>

%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1006/jnth.1994.1073">On Lattices Equivalent to Their Duals</a>, J. Number Theory, 48 (1994), 373-382.

%H K. L. Fields, <a href="http://dx.doi.org/10.1107/S0567739480000411">The fragile lattice packings of spheres in three-dimensional space</a>, Acta Cryst. Sect. A 36 (1980), 194-197.

%H A. Patterson, <a href="http://dx.doi.org/10.1063/1.1769865">Crystal lattice models based on the close packing of spheres</a>, Rev. Sci. Instrumen., 12 (1941), 206-211.

%e 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 + ...

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

%Y Cf. A028966.

%K nonn

%O 0,3

%A _N. J. A. Sloane_.

%E Edited by _N. J. A. Sloane_, Sep 29 2006

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)