A008391 Coordination sequence for A_8 lattice.

1, 72, 1332, 11832, 66222, 271224, 889716, 2476296, 6077196, 13507416, 27717948, 53265960, 96900810, 168278760, 280819260, 452715672, 708113304, 1078467624, 1604095524, 2335932504, 3337508646, 4687156248, 6480461988, 8832976488, 11883194148, 15795816120

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.

R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

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 (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = n*(715*n^6 + 6006*n^4 + 10395*n^2 + 3044)/280, a(0) = 1.

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Mar 04 2012

G.f.: (1 + 64*x + 784*x^2 + 3136*x^3 + 4900*x^4 + 3136*x^5 + 784*x^6 + 64*x^7 + x^8)/(1-x)^8. - Colin Barker, Sep 26 2012

Maple

1, seq(n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280, n=1..40);

Mathematica

Join[{1}, Table[143/56n^7+429/20n^5+297/8n^3+761/70n, {n, 30}]] (* or *)
Join[{1}, LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {72, 1332, 11832, 66222, 271224, 889716, 2476296, 6077196}, 30]] (* Harvey P. Dale, Mar 04 2012 *)

Prog

(Maxima)
a[0]:1$
a[1]:72$
a[2]:1332$
a[3]:11832$
a[4]:66222$
a[5]:271224$
a[6]:889716$
a[7]:2476296$
a[8]:6077196$
a[n]:=8*a[n-1]-28*a[n-2]+ 56*a[n-3]- 70*a[n-4]+56*a[n-5]-28*a[n-6]+8*a[n-7]-a[n-8];
makelist(a[n], n, 0, 30); /* Martin Ettl, Oct 26 2012 */
(Magma) [1] cat [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280: n in [1..40]]; // G. C. Greubel, May 26 2023
(SageMath) [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)//280 +int(n==0) for n in range(41)] # G. C. Greubel, May 26 2023

Crossrefs

Row 8 of A103881.

Sequence in context: A239423 A128800 A367781 * A292881 A282018 A037251

Adjacent sequences: A008388 A008389 A008390 * A008392 A008393 A008394

Keyword

nonn,easy

Author

N. J. A. Sloane and J. H. Conway

Status

approved