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

Author

Roger L. Bagula, Aug 12 2006

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A180250.

Programs

Magma

R:=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

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 *)

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

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

Extensions

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

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