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%I #13 Dec 24 2015 14:10:55
%S 0,1,1,2,5,19,84,393,1865,8886,42381,202187,964640,4602409,21958729,
%T 104768258,499864605,2384926971,11378834836,54290082897,259025915025,
%U 1235850473974,5896423120549,28132695944723,134225201438720
%N Expansion of x*(1-4*x-3*x^2)/(1-5*x+5*x^3+x^4).
%C Sequence produced by 4 X 4 Markov chain with symmetric quartic characteristic polynomial x^4-5*x^3+5*x+1.
%C Setting m=3 gives a Fibonacci sequence.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,0,-5,-1).
%F 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]].
%F 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
%t 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}]
%t 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 *)
%o (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
%Y Cf. A107378.
%K nonn,easy
%O 0,4
%A _Roger L. Bagula_, May 24 2005
%E Edited by _N. J. A. Sloane_, Jul 13 2007
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