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

0,2

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.

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

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

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 *)

Prog

(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
(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
(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

Crossrefs

Sequence in context: A305549 A320214 A346642 * A090685 A232297 A008392

Adjacent sequences: A008397 A008398 A008399 * A008401 A008402 A008403

Keyword

nonn,easy,nice

Author

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

Status

approved