A019558 - OEIS

A019558

Coordination sequence for F_4 lattice.

1, 48, 384, 1392, 3456, 6960, 12288, 19824, 29952, 43056, 59520, 79728, 104064, 132912, 166656, 205680, 250368, 301104, 358272, 422256, 493440, 572208, 658944, 754032, 857856, 970800, 1093248, 1225584, 1368192, 1521456, 1685760, 1861488, 2049024, 2248752

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Michael Baake and Uwe Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.

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

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = 16*n*(2 - 3*n + 4*n^2), n >= 1.

G.f.: (1 + 44*x + 198*x^2 + 140*x^3 + x^4)/(1-x)^4 = 1 + 48*x*(x+1)*(3*x+1)/(x-1)^4.

From Elmo R. Oliveira, Sep 06 2025: (Start)

E.g.f.: 1 + 16*exp(x)*x*(3 + 9*x + 4*x^2).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.

a(n) = 3*A167471(n) for n > 0. (End)

MATHEMATICA

Join[{1}, Table[16n (2-3n+4n^2), {n, 30}]] (* Harvey P. Dale, Oct 21 2011 *)

PROG

(Magma) [1] cat [16*n*(2-3*n+4*n^2): n in [1..50]]; // Vincenzo Librandi, Oct 22 2011