A008402 Crystal ball sequence for {E_6}* lattice.

1, 55, 883, 6085, 26461, 86491, 232975, 545833, 1151065, 2235871, 4065931, 7004845, 11535733, 18284995, 28048231, 41818321, 60815665, 86520583, 120707875, 165483541, 223323661, 297115435

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

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. [Annotated scanned copy]

Index entries for crystal ball sequences

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

G.f.: (1 + 48*x + 519*x^2 + 1024*x^3 + 519*x^4 + 48*x^5 + x^6)/(1-x)^7.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=1, a(1)=55, a(2)=883, a(3)=6085, a(4)=26461, a(5)=86491, a(6)=232975. - Harvey P. Dale, Jun 20 2013

a(n) = 3*n^6 + 9*n^5 + 15*n^4 + 15*n^3 + 9*n^2 + 3*n + 1 = 1 + 3*n*(n+1)*(n^2+n+1)^2. - Charles R Greathouse IV, Jun 20 2013

E.g.f.: exp(x)*(1 + 54*x + 387*x^2 + 600*x^3 + 300*x^4 + 54*x^5 + 3*x^6). - Stefano Spezia, Apr 15 2022

Mathematica

CoefficientList[Series[(1+48x+519x^2+1024x^3+519x^4+48x^5+x^6)/(1-x)^7, {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 55, 883, 6085, 26461, 86491, 232975}, 30] (* Harvey P. Dale, Jun 20 2013 *)

Prog

(PARI) 3*n^6+9*n^5+15*n^4+15*n^3+9*n^2+3*n+1 \\ Charles R Greathouse IV, Jun 20 2013
(Magma) [1 +3*n*(n+1)*(n^2+n+1)^2: n in [0..40]]; // G. C. Greubel, May 31 2023
(SageMath) [1 +3*n*(n+1)*(n^2+n+1)^2 for n in range(41)] # G. C. Greubel, May 31 2023

Crossrefs

Partial sums of A008401.

Sequence in context: A221786 A358634 A241634 * A281073 A329753 A203872

Adjacent sequences: A008399 A008400 A008401 * A008403 A008404 A008405

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved