A107377 Expansion of x*(1-4*x-3*x^2)/(1-5*x+5*x^3+x^4).

0, 1, 1, 2, 5, 19, 84, 393, 1865, 8886, 42381, 202187, 964640, 4602409, 21958729, 104768258, 499864605, 2384926971, 11378834836, 54290082897, 259025915025, 1235850473974, 5896423120549, 28132695944723, 134225201438720

Offset

0,4

Comments

Sequence produced by 4 X 4 Markov chain with symmetric quartic characteristic polynomial x^4-5*x^3+5*x+1.

Setting m=3 gives a Fibonacci sequence.

Links

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (5,0,-5,-1).

Formula

Let m=5, M={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-1, -m, 0, m}}, v[n]=M.v[n-1], then a(n) = v[n][[1]].

a(0)=0, a(1)=1, a(2)=1, a(3)=2, a(n)=5*a(n-1)-5*a(n-3)-a(n-4). - Harvey P. Dale, Dec 24 2015

Mathematica

m = 5 M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-1, -m, 0, m}} Expand[Det[M - x*IdentityMatrix[4]]] NSolve[Det[M - x*IdentityMatrix[4]] == 0, x] v[1] = {0, 1, 1, 2}; v[n_] := v[n] = M.v[n - 1]; digits = 50; a = Table[v[n][[1]], {n, 1, digits}]
CoefficientList[Series[x (1-4x-3x^2)/(1-5x+5x^3+x^4), {x, 0, 30}], x] (* or *) LinearRecurrence[{5, 0, -5, -1}, {0, 1, 1, 2}, 30] (* Harvey P. Dale, Dec 24 2015 *)

Prog

(PARI) Vec(x*(1-4*x-3*x^2)/(1-5*x+5*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Crossrefs

Cf. A107378.

Sequence in context: A262165 A219661 A179566 * A286886 A058132 A286071

Adjacent sequences: A107374 A107375 A107376 * A107378 A107379 A107380

Keyword

nonn,easy

Author

Roger L. Bagula, May 24 2005

Extensions

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

Status

approved