A008349 Crystal ball sequence for E_8 lattice. (Formerly M5419)

1, 241, 9361, 131041, 996001, 5109841, 20015281, 64495681, 179375041, 444798001, 1006201681, 2111519521, 4162485601, 7783236241, 13909734001, 23903867521, 39696408961, 63963339121, 100340378641, 153680892001

Offset

0,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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 (9, -36, 84, -126, 126, -84, 36, -9, 1).

Formula

G.f.: (1 + 232*x + 7228*x^2 + 55384*x^3 + 133510*x^4 + 107224*x^5 + 24508*x^6 + 232*x^7 + x^8)/(1 - x)^9.

a(0)=1, a(1)=241, a(2)=9361, a(3)=131041, a(4)=996001, a(5)=5109841, a(6)=20015281, a(7)=64495681, a(8)=179375041, a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - Harvey P. Dale, Jun 12 2012

Maple

57/7*n^8+108/7*n^7+30*n^6+72*n^5+39*n^4+36*n^3+300/7*n^2-24/7*n+1;

Mathematica

CoefficientList[Series[(1+232x+7228x^2+107224x^5+133510x^4+ 55384x^3+ 24508x^6+ 232x^7+ x^8)/(1-x)^9, {x, 0, 30}], x] (* or *) LinearRecurrence[ {9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 241, 9361, 131041, 996001, 5109841, 20015281, 64495681, 179375041}, 30] (* Harvey P. Dale, Jun 12 2012 *)

Prog

(Python)
A008349_list, m = [], [328320, -1071360, 1347840, -812160, 233280, -25920, 240, 0, 1]
for _ in range(10**2):
    A008349_list.append(m[-1])
    for i in range(8):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015
(Magma) [57/7*n^8 + 108/7*n^7 + 30*n^6 + 72*n^5 + 39*n^4 + 36*n^3 + 300/7*n^2 - 24/7*n + 1: n in [0..40]]; // Vincenzo Librandi, Dec 16 2015
(PARI) a(n)=(57*n^8 + 108*n^7 + 210*n^6 + 504*n^5 + 273*n^4 + 252*n^3 + 300*n^2 - 24*n + 7)/7 \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Sequence in context: A165375 A183887 A007205 * A094732 A119728 A133327

Adjacent sequences: A008346 A008347 A008348 * A008350 A008351 A008352

Keyword

nonn,easy,nice

Author

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

Extensions

The values given by O'Keeffe are incorrect.

Status

approved