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Coordination sequence for 8-dimensional cubic lattice.
5

%I #70 Nov 08 2024 04:36:40

%S 1,16,128,688,2816,9424,27008,68464,157184,332688,658048,1229360,

%T 2187520,3732560,6140800,9785072,15158272,22900496,33830016,48978352,

%U 69629696,97364944,134110592,182192752,244396544,324031120,425000576

%N Coordination sequence for 8-dimensional cubic lattice.

%C Coordination sequence for 8-dimensional cyclotomic lattice Z[zeta_16].

%H Seiichi Manyama, <a href="/A008416/b008416.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H M. Beck and S. Hosten, <a href="https://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.

%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H Ross McPhedran, <a href="https://arxiv.org/abs/2311.06294">Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis</a>, arXiv:2311.06294 [math.NT], 2023. See p. 6.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 56, -70, 56, -28, 8, -1).

%F G.f.: ((1+x)/(1-x))^8.

%F a(n) = A008415(n) + A008415(n-1) + a(n-1). - _Bruce J. Nicholson_, Dec 17 2017

%F n*a(n) = 16*a(n-1) + (n-2)*a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018

%t CoefficientList[Series[((1 + x)/(1 - x))^8, {x, 0, 26}], x] (* _Michael De Vlieger_, Dec 18 2017 *)

%Y Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008418, A008420.

%Y Cf. A113413, A122542.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_