|
%I #96 Mar 14 2024 16:41:13
%S 1,8,32,88,192,360,608,952,1408,1992,2720,3608,4672,5928,7392,9080,
%T 11008,13192,15648,18392,21440,24808,28512,32568,36992,41800,47008,
%U 52632,58688,65192,72160,79608,87552,96008,104992,114520,124608,135272
%N Coordination sequence for 4-dimensional cubic lattice (points on surface of 4-dimensional cross-polytope).
%C Coordination sequence for 4-dimensional cyclotomic lattice Z[zeta_8].
%C If Y_i (i=1,2,3,4) are 2-blocks of a (n+4)-set X then a(n-3) is the number of 7-subsets of X intersecting each Y_i (i=1,2,3,4). - _Milan Janjic_, Oct 28 2007
%H Seiichi Manyama, <a href="/A008412/b008412.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 Pierre de la Harpe, <a href="https://arxiv.org/abs/2106.02499">On the prehistory of growth of groups</a>, arXiv:2106.02499 [math.GR], 2021.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%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 M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.
%H M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: ((1+x)/(1-x))^4.
%F a(n) = 8*n*(n^2+2)/3 for n>1.
%F a(n) = 8*A006527(n) for n>0.
%F a(n) = A005899(n) + A005899(n-1) + a(n-1). - _Bruce J. Nicholson_, Dec 17 2017
%F n*a(n) = 8*a(n-1) + (n-2)*a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018
%F a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=4, for n>=1. - _Shel Kaphan_, Feb 14 2023
%F a(n) = A035598(n)*4/n, for n>0. - _Shel Kaphan_, Feb 28 2023
%F E.g.f.: 1 + 8*exp(x)*x*(3 + 3*x + x^2)/3. - _Stefano Spezia_, Mar 14 2024
%p 8/3*n^3+16/3*n;
%t CoefficientList[Series[((1+x)/(1-x))^4,{x,0,40}],x] (* or *)
%t LinearRecurrence[{4, -6, 4, -1}, {1, 8, 32, 88, 192}, 41] (* _Harvey P. Dale_, Jun 10 2011 *)
%t f[n_] := 8 n (n^2 + 2)/3; f[0] = 1; Array[f, 38, 0] (* or *)
%t g[n_] := 4n^2 +2; f[n_] := f[n-1] + g[n] + g[n -1]; f[0] = 1; f[1] = 8; Array[f, 38, 0] (* _Robert G. Wilson v_, Dec 27 2017 *)
%o (PARI) a(n)=if(n,8*(n^2+2)*n/3,1) \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Magma) I:=[1,8,32,88,192]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 15 2018
%Y Cf. A001845, A005899, A006527, A035598.
%Y First differences of A001846.
%Y Row 4 of A035607, A266213.
%Y Column 4 of A113413, A119800, A122542.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
|
