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A008412 Coordination sequence for 4-dimensional cubic lattice (points on surface of 4-dimensional cross-polytope). 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

Milan Janjic, Two Enumerative Functions

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).

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

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 *)

PROG

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

(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

CROSSREFS

Cf. A001845, A005899.

First differences of A001846.

Sequence in context: A211633 A130809 A018839 * A014819 A033155 A132117

Adjacent sequences:  A008409 A008410 A008411 * A008413 A008414 A008415

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 19 09:28 EDT 2018. Contains 316339 sequences. (Running on oeis4.)