A008412 Coordination sequence for 4-dimensional cubic lattice (points on surface of 4-dimensional cross-polytope).

1, 8, 32, 88, 192, 360, 608, 952, 1408, 1992, 2720, 3608, 4672, 5928, 7392, 9080, 11008, 13192, 15648, 18392, 21440, 24808, 28512, 32568, 36992, 41800, 47008, 52632, 58688, 65192, 72160, 79608, 87552, 96008, 104992, 114520, 124608, 135272

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Coordination sequence for 4-dimensional cyclotomic lattice Z[zeta_8].

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
• Milan Janjic, Two Enumerative Functions
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
• Ross McPhedran, Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis, arXiv:2311.06294 [math.NT], 2023. See p. 6.
• M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
• M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A001845, A005899, A006527, A035598.

First differences of A001846.

Row 4 of A035607, A266213.

Column 4 of A113413, A119800, A122542.

Programs

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

Maple

8/3*n^3+16/3*n;

Mathematica

CoefficientList[Series[((1+x)/(1-x))^4,{x,0,40}],x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 8, 32, 88, 192}, 41] (* Harvey P. Dale, Jun 10 2011 *)
f[n_] := 8 n (n^2 + 2)/3; f[0] = 1; Array[f, 38, 0] (* or *)
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 *)

PARI

a(n)=if(n,8*(n^2+2)*n/3,1) \\ Charles R Greathouse IV, Jun 10 2011

Formula

G.f.: ((1+x)/(1-x))^4.

a(n) = 8*n*(n^2+2)/3 for n>1.

a(n) = 8*A006527(n) for n>0.

a(n) = A005899(n) + A005899(n-1) + a(n-1). - Bruce J. Nicholson, Dec 17 2017

n*a(n) = 8*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=4, for n>=1. - Shel Kaphan, Feb 14 2023

a(n) = A035598(n)*4/n, for n>0. - Shel Kaphan, Feb 28 2023

E.g.f.: 1 + 8*exp(x)*x*(3 + 3*x + x^2)/3. - Stefano Spezia, Mar 14 2024