A008397 Coordination sequence for E_7 lattice.

1, 126, 2898, 25886, 133506, 490014, 1433810, 3573054, 7902594, 15942206, 29896146, 52834014, 88892930, 143501022, 223622226, 338022398, 497556738, 715478526, 1007769170, 1393489566

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 (7,-21,35,-35,21,-7,1).

Formula

a(n) = (2/5)*(74*n^6 - 6*n^5 + 130*n^4 + 30*n^3 + 106*n^2 - 24*n + 5) for n >= 1.

Bacher et al. give a g.f.

G.f.: (1 + 119*x + 2037*x^2 + 8211*x^3 + 8787*x^4 + 2037*x^5 + 119*x^6 + x^7)/(1-x)^7. - Colin Barker, Sep 26 2012

Maple

a:= n-> `if`(n=0, 1, 148/5*n^6-12/5*n^5+52*n^4+12*n^3+212/5*n^2-48/5*n+2):
seq(a(n), n=0..25);

Mathematica

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 126, 2898, 25886, 133506, 490014, 1433810, 3573054}, 20] (* Harvey P. Dale, Nov 12 2014 *)

Prog

(Magma) [1] cat [(2/5)*(74*n^6 -6*n^5 +130*n^4 +30*n^3 +106*n^2 -24*n + 5): n in [1..30]]; // G. C. Greubel, May 29 2023
(SageMath) [2*(74*n^6 -6*n^5 +130*n^4 +30*n^3 +106*n^2 -24*n +5)//5 - int(n==0) for n in range(31)] # G. C. Greubel, May 29 2023

Crossrefs

Sequence in context: A202398 A113857 A267282 * A329754 A202593 A293104

Adjacent sequences: A008394 A008395 A008396 * A008398 A008399 A008400

Keyword

nonn,easy

Author

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

Status

approved