A120717 Expansion of x*(67 + 134*x - 287*x^2 - 378*x^3)/((1+2*x)*(1-3*x)*(1 - 5*x - 10*x^2)).

0, 67, 536, 3666, 24834, 163870, 1077594, 7054814, 46137578, 301492462, 1969619930, 12865344702, 84029934282, 548824052494, 3584482608186, 23410842173150, 152899603572266, 998608137861166, 6522041823044762

Offset

0,2

Comments

Old name was: Sequence produced by Markov chain based on body-centered pentagonal prism 11 X 11 bond graph.

Characteristic polynomial = -480*x^4 - 1040*x^5 - 752*x^6 - 120*x^7 + 90*x^8 + 35*x^9 - x^11.

The molecule with this structure is known as Ferrocene.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Dr Karl Harrison, Ferrocene, What is Ferrocene? About its Science, Chemistry and Structure

Wikipedia, Ferrocene

Index entries for linear recurrences with constant coefficients, signature (6,11,-40,-60).

Formula

G.f.: x*(67 + 134*x - 287*x^2 - 378*x^3)/((1+2*x)*(1-3*x)*(1 - 5*x - 10*x^2)). - Colin Barker, Nov 29 2012

a(n) = (1/10)*(-63*[n=0] + 49*(-2)^n - 74*3^n + 22*(4*A180250(n) + 25*A180250(n-1))). - G. C. Greubel, Jul 21 2023

Mathematica

M0 = {{0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1}, {1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1}, {1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1}, {0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1}, {0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1}, {0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0}}; v[1]= Table[Fibonacci[n], {n, 0, 10}]; v[n_]:= v[n] = M0.v[n-1]; Table[v[n][[1]], {n, 50}]
LinearRecurrence[{6, 11, -40, -60}, {0, 67, 536, 3666, 24834}, 20] (* Harvey P. Dale, May 25 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(67+134*x-287*x^2-378*x^3)/((1+2*x)*(1-3*x)*(1-5*x-10*x^2)) )); // G. C. Greubel, Jul 21 2023
(SageMath)
A180250= BinaryRecurrenceSequence(5, 10, 0, 1)
def A120717(n): return (1/10)*(-63*int(n==0) + 49*(-2)^n - 74*3^n + 22*(4*A180250(n+1) + 25*A180250(n)))
[A120717(n) for n in range(41)] # G. C. Greubel, Jul 21 2023

Crossrefs

Cf. A180250.

Sequence in context: A219736 A257747 A163150 * A200963 A245878 A289867

Adjacent sequences: A120714 A120715 A120716 * A120718 A120719 A120720

Keyword

nonn,easy

Author

Roger L. Bagula, Aug 12 2006

Extensions

Edited by N. J. A. Sloane, Jul 13 2007

New name (using g.f. by Colin Barker) by Joerg Arndt, Jan 07 2013

Status

approved