A007900 Coordination sequence for D_4 lattice.

1, 24, 144, 456, 1056, 2040, 3504, 5544, 8256, 11736, 16080, 21384, 27744, 35256, 44016, 54120, 65664, 78744, 93456, 109896, 128160, 148344, 170544, 194856, 221376, 250200, 281424, 315144, 351456, 390456, 432240, 476904, 524544, 575256, 629136, 686280, 746784, 810744

Offset

0,2

References

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

Links

Paolo Xausa, 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, 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, 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(12) (1995), 905-908.

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210(12) (1995), 905-908. [Annotated scanned copy]

Index entries for sequences related to D_4 lattice

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

Formula

G.f.: (1+54*x^2+20*x+20*x^3+x^4)/(1-x)^4 = 1+24*x*(x+1)^2/(x-1)^4.

G.f. for coordination sequence of D_n lattice: (Sum(binomial(2*n, 2*i)*z^i, i=0..n)-2*n*z*(1+z)^(n-2))/(1-z)^n.

Maple

if n=0 then 1 else 8*n*(2*n^2+1); fi;

Mathematica

A007900[n_] := If[n == 0, 1, 8*n*(2*n^2 + 1)]; Array[A007900, 50, 0] (* Paolo Xausa, May 20 2026 *)
(* Alternative: *)
LinearRecurrence[{4, -6, 4, -1}, {1, 24, 144, 456, 1056}, 50] (* Paolo Xausa, May 20 2026 *)

Crossrefs

A row of array A103903.

Sequence in context: A326856 A358960 A076835 * A158874 A059593 A200194

Adjacent sequences: A007897 A007898 A007899 * A007901 A007902 A007903

Keyword

nonn,easy

Author

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

Extensions

More terms from Paolo Xausa, May 20 2026

Status

approved