A008401 Coordination sequence for {E_6}* lattice.

1, 54, 828, 5202, 20376, 60030, 146484, 312858, 605232, 1084806, 1830060, 2938914, 4530888, 6749262, 9763236, 13770090, 18997344, 25704918, 34187292, 44775666, 57840120, 73791774, 93084948

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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
• 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).

Crossrefs

Cf. A008402 (partial sums).

Programs

Magma

[1] cat [6*n*(3*n^4+5*n^2+1): n in [1..40]]; // G. C. Greubel, May 30 2023

Mathematica

Join[{1},Table[18 n^5+30 n^3+6 n,{n,30}]] (* Harvey P. Dale, May 16 2012 *)

SageMath

[6*n*(3*n^4+5*n^2+1) +int(n==0) for n in range(41)] # G. C. Greubel, May 30 2023

Formula

a(n) = 6*n*(3*n^4 + 5*n^2 + 1), n > 0.

G.f.: (1+48*x+519*x^2+1024*x^3+519*x^4+48*x^5+x^6)/(1-x)^6.

E.g.f.: 1 + 6*exp(x)*x*(9 + 60*x + 80*x^2 + 30*x^3 + 3*x^4). - Stefano Spezia, Apr 15 2022