A008399 Coordination sequence for E_6 lattice.

1, 72, 1062, 6696, 26316, 77688, 189810, 405720, 785304, 1408104, 2376126, 3816648, 5885028, 8767512, 12684042, 17891064, 24684336, 33401736, 44426070, 58187880, 75168252, 95901624, 120978594, 151048728, 186823368, 229078440, 278657262, 336473352, 403513236

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 (6,-15,20,-15,6,-1).

Formula

a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.

Bacher et al. give a g.f.

G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^6 = 1 + 18*x*(4+35*x+78*x^2+35*x^3+4*x^4)/(1-x)^6. - Colin Barker, Sep 26 2012

E.g.f.: 1 + (1/5)*x*(360 + 2295*x + 3105*x^2 + 1170*x^3 + 117*x^4 )*exp(x). - G. C. Greubel, May 29 2023

Maple

1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 72, 1062, 6696, 26316, 77688, 189810}, 30] (* Harvey P. Dale, Oct 24 2022 *)

Prog

(Magma) [1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // G. C. Greubel, May 29 2023
(SageMath) [9*n*(13*n^2+7)*(n^2+1)//5 +int(n==0) for n in range(41)] # G. C. Greubel, May 29 2023

Crossrefs

Cf. A008397, A008340, A019557, A019558.

Sequence in context: A019562 A250423 A250650 * A232573 A022148 A192847

Adjacent sequences: A008396 A008397 A008398 * A008400 A008401 A008402

Keyword

nonn,easy

Author

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

Status

approved