A008383 Coordination sequence for A_4 lattice.

1, 20, 110, 340, 780, 1500, 2570, 4060, 6040, 8580, 11750, 15620, 20260, 25740, 32130, 39500, 47920, 57460, 68190, 80180, 93500, 108220, 124410, 142140, 161480, 182500, 205270, 229860, 256340, 284780, 315250, 347820, 382560, 419540, 458830, 500500, 544620

Offset

0,2

References

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

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).

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]

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

Formula

a(n) = 5*n*(7*n^2 + 5)/3, a(0) = 1.

G.f.: (1+16*x+36*x^2+16*x^3+x^4)/(1-x)^4 = 1+10*x*(2+3*x+2*x^2)/(x-1)^4. - Colin Barker, Apr 13 2012

E.g.f.: (1/3)*(3 + 5*x*(12 + 21*x + 7*x^2)*exp(x)). - G. C. Greubel, May 25 2023

Maple

a:= n-> `if`(n=0, 1, 35/3*n^3+25/3*n): seq (a(n), n=0..50);

Mathematica

CoefficientList[Series[(1+16x+36x^2+16x^3+x^4)/(1-x)^4, {x, 0, 40}], x] (* Harvey P. Dale, Dec 01 2013 *)
Join[{1}, LinearRecurrence[{4, -6, 4, -1}, {20, 110, 340, 780}, 40]] (* Jean-François Alcover, Jan 07 2019 *)

Prog

(Magma) [n eq 0 select 1 else 5*n*(7*n^2+5)/3: n in [0..45]]; // G. C. Greubel, May 25 2023
(SageMath) [5*n*(7*n^2+5)/3+int(n==0) for n in range(46)] # G. C. Greubel, May 25 2023

Crossrefs

Cf. A005901, A008384, A008385.

Sequence in context: A189437 A285698 A109854 * A024192 A309781 A228243

Adjacent sequences: A008380 A008381 A008382 * A008384 A008385 A008386

Keyword

nonn,easy

Author

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

Status

approved