A008361 Coordination sequence for D_8 lattice.

1, 112, 2592, 25424, 149568, 629808, 2100832, 5910288, 14610560, 32641008, 67232416, 129565392, 236214464, 410909616, 686647008, 1108180624, 1734926592, 2644311920, 3935599392, 5734220368

Offset: 0

Author

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

Keywords

• nonn
• nice
• easy

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, 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).

Crossrefs

A row of array A103903.

Programs

Maple

1984/315*n^7+1888/45*n^5+2368/45*n^3+1168/105*n;

Mathematica

Join[{1},Table[16/315*n*(219+1036*n^2+826*n^4+124*n^6),{n,30}]] (* or *) Join[{1},LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{112,2592,25424,149568,629808,2100832,5910288,14610560},30]] (* Harvey P. Dale, Feb 21 2012 *)

PARI

a(n) = 16/315*n*(219 + 1036*n^2 + 826*n^4 + 124*n^6) \\ Charles R Greathouse IV, Feb 10 2017

Formula

a(n) = 16/315*n*(219 + 1036*n^2 + 826*n^4 + 124*n^6). - Harvey P. Dale, Feb 21 2012

a(0)=1, a(1)=112, a(2)=2592, a(3)=25424, a(4)=149568, a(5)=629808, a(6)=2100832, a(7)=5910288, a(8)=14610560, 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, Feb 21 2012

G.f.: (x^8 + 104*x^7 + 1724*x^6 + 7768*x^5 + 12550*x^4 + 7768*x^3 + 1724*x^2 + 104*x + 1)/(x - 1)^8. - Colin Barker, Sep 26 2012