A008413 Coordination sequence for 5-dimensional cubic lattice.

1, 10, 50, 170, 450, 1002, 1970, 3530, 5890, 9290, 14002, 20330, 28610, 39210, 52530, 69002, 89090, 113290, 142130, 176170, 216002, 262250, 315570, 376650, 446210, 525002, 613810, 713450, 824770, 948650, 1086002, 1237770, 1404930

Offset

0,2

Comments

If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Oct 28 2007

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Milan Janjic, Two Enumerative Functions

Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: ((1+x)/(1-x))^5.

a(n) = (4/3)*n^4 + (20/3)*n^2 + 2 for n > 0. - Michael De Vlieger, Oct 04 2016

n*a(n) = 10*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

From Shel Kaphan, Mar 03 2023: (Start)

a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=5, for n>=1.

a(n) = A035599(n)*5/n, for n>0. (End)

Maple

4/3*n^4+20/3*n^2+2;

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 50, 170, 450, 1002}, 40] (* Harvey P. Dale, May 02 2016 *)
{1}~Join~Table[4/3 n^4 + 20/3 n^2 + 2, {n, 32}] (* or *)
CoefficientList[Series[((1 + x)/(1 - x))^5, {x, 0, 32}], x] (* Michael De Vlieger, Oct 04 2016 *)

Crossrefs

Cf. A035599.

Row 5 of A035607, A266213.

Column 5 of A113413, A119800, A122542.

Sequence in context: A337732 A372496 A051230 * A006542 A237655 A261648

Adjacent sequences: A008410 A008411 A008412 * A008414 A008415 A008416

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved