%I #13 Aug 30 2020 15:23:22

%S 0,1,1,2,4,11,35,122,440,1609,5913,21770,80204,295555,1089227,4014322,

%T 14794864,54526993,200961457,740652050,2729705364,10060448635,

%U 37078224883,136653426026,503642204200,1856195468633,6841089945545

%N Expansion of x*(1-3*x-2*x^2)/(1-4*x+4*x^3+x^4).

%C Sequence produced by 4 X 4 Markov chain with characteristic polynomial x^4-4*x^3+4*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 (4,0,-4,-1)

%t m = 4 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 (1-3x-2x^2)/(1-4x+4x^3+x^4),{x,0,30}],x] (* or *) LinearRecurrence[{4,0,-4,-1},{0,1,1,2},30] (* _Harvey P. Dale_, Aug 30 2020 *)

%o (PARI) Vec(x*(1-3*x-2*x^2)/(1-4*x+4*x^3+x^4)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%Y Cf. A107377.

%K nonn,easy

%O 0,4

%A _Roger L. Bagula_, May 24 2005

%E Edited by _N. J. A. Sloane_, Jul 13 2007