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A005886 Theta series of f.c.c. lattice with respect to tetrahedral hole.

%I A005886 M3429 #42 Feb 21 2018 07:18:49
%S A005886 4,12,12,16,24,12,24,36,12,28,36,24,36,36,24,24,60,36,28,48,12,60,60,
%T A005886 24,48,48,36,48,60,24,52,84,48,24,60,36,48,96,36,72,48,36,72,60,48,52,
%U A005886 96,36,60,96,24,72,108,24,48,60,72,96,84,60,48,108,36,52,72,60,108,108,36,48,108
%N A005886 Theta series of f.c.c. lattice with respect to tetrahedral hole.
%C A005886 Empirically, the number of integral quadruples with sum = 1, sum-of-squares = 2n-1. - _Colin Mallows_, Dec 31 2016
%D A005886 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005886 G. C. Greubel, <a href="/A005886/b005886.txt">Table of n, a(n) for n = 0..1000</a>
%H A005886 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html">Home page for this lattice</a>
%H A005886 N. J. A. Sloane and B. K. Teo, <a href="http://dx.doi.org/10.1063/1.449551">Theta series and magic numbers for close-packed spherical clusters</a>, J. Chem. Phys. 83 (1985) 6520-6534.
%H A005886 <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>
%F A005886 a(n) = 1/2 * A005878(n) = 2 * A005869(n) = 4 * A008443(n). - _Michael Somos_, May 31 2012
%e A005886 4 + 12*x + 12*x^2 + 16*x^3 + 24*x^4 + 12*x^5 + 24*x^6 + 36*x^7 + 12*x^8 + ...
%t A005886 QP = QPochhammer; CoefficientList[4(QP[q^2]^2/QP[q])^3 + O[q]^50, q] (* _Jean-François Alcover_, Jul 04 2017, after _Michael Somos_ *)
%Y A005886 Cf. A005869, A005878, A008443. Partial sums is A121054. Cf also A278081-A278086.
%K A005886 nonn,easy
%O A005886 0,1
%A A005886 _N. J. A. Sloane_
%E A005886 Terms a(50) onward added by _G. C. Greubel_, Feb 20 2018