A008400 Crystal ball sequence for E_6 lattice.

1, 73, 1135, 7831, 34147, 111835, 301645, 707365, 1492669, 2900773, 5276899, 9093547, 14978575, 23746087, 36430129, 54321193, 79005529, 112407265, 156833335, 215021215, 290189467, 386091091

Offset: 0

Author

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

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

Programs

Magma

[1 +3*n*(n+1)*(26*n^4 +52*n^3 +73*n^2 +47*n +42)/20: n in [0..40]]; // G. C. Greubel, May 30 2023

Maple

seq(39/10*n^6+117/10*n^5+75/4*n^4+18*n^3+267/20*n^2+63/10*n+1, n=0..35);

Mathematica

Table[39/10 n^6+117/10 n^5+75/4 n^4+18n^3+267/20 n^2+63/10 n+1, {n,0, 30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,73,1135, 7831,34147,111835,301645},30] (* Harvey P. Dale, Feb 23 2015 *)

PARI

a(n)=(78*n^6 + 234*n^5 + 375*n^4 + 360*n^3 + 267*n^2 + 126*n + 20)/20 \\ Charles R Greathouse IV, Feb 10 2017

SageMath

[1 +3*n*(n+1)*(26*n^4 +52*n^3 +73*n^2 +47*n +42)//20 for n in range(41)] # G. C. Greubel, May 30 2023

Formula

a(n) = 1 + (3/20)*n*(n+1)*(26*n^4 + 52*n^3 + 73*n^2 + 47*n + 42).

G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^7. - Colin Barker, Mar 16 2012

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). - Harvey P. Dale, Feb 23 2015

E.g.f.: (1/20)*(20 + 1440*x + 9900*x^2 + 15480*x^3 + 7785*x^4 + 1404*x^5 + 78*x^6)*exp(x). - G. C. Greubel, May 30 2023