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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Crossrefs

Partial sums of A008401.

Programs

Magma

[1 +3*n*(n+1)*(n^2+n+1)^2: n in [0..40]]; // G. C. Greubel, May 31 2023

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 *)

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

SageMath

[1 +3*n*(n+1)*(n^2+n+1)^2 for n in range(41)] # G. C. Greubel, May 31 2023

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