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The sequence is an expansion based on a Padovan polynomial inside a Fibonacci polynomial (a Pisot): 2 - 3 x + 4 x^2 - x^3.
a(n) = 4a4*a(n-1) - 3a3*a(n-2) + 2a2*a(n-3), starting 0, 1, 4.
(* expansion form: toral inverse of the derived polynomial*);
f[x_] = ExpandAll[4*(x - (1 - Sqrt[
x^3 - x - 1])/2)*(x - (1 + Sqrt[x^3 - x - 1])/2)];
g[x_] = ExpandAll[x^3*f[1/x]];
a = Table[SeriesCoefficient[
Series[-1/g[x], {x, 0, 50}], n], {n, 0, 50}]
(* vector Matrix Markov version derived*);
CompanionMatrix[p_, x_] := Module[{cl = CoefficientList[p, x], deg,
m}, cl = Drop[cl/Last[cl], -1]; deg = Length[cl];
If[deg == 1, {-cl}, m = RotateLeft[
IdentityMatrix[deg]]; m[[ -1]] = -cl; Transpose[m]]];
M = Transpose[CompanionMatrix[f[x], x]];
v[0] = Table[a[[n]], {n, 1, 3}]
v[n_] := v[n] = M.v[n - 1];
Table[v[n][[1]], {n, 0, 50}]
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<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (4,-3,2).
<a href="/index/Rec">Index entries for to sequences related to with linear recurrences with constant coefficients</a>, signature (4,-3,2).
<a href="/index/Rea#recLCCRec">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (4,-3,2).
_Roger L. Bagula_, Dec 02 2010
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(PARI) Vec(x/(1-4*x+3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
nonn,easy
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editing
_Roger Bagula (rlbagulatftn(AT)yahoo.com), _, Dec 02 2010
<a href="/Sindx_index/Rea.html#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (4,-3,2).