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This can also be defined as the expansion of 1/(x^4*p(1/x))) with p (x) = 2 - 2*x + x^2 - 3*x^3 + x^4.

Limit _{n -> infinity oo} a(n+1)/a(n) approaches the Pisot root 2.8071578467023431323785220673259635911...

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<a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,2,-2).

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

1, 3, 8, 23, 65, 182, 511, 1435, 4028, 11307, 31741, 89102, 250123, 702135, 1971004, 5532919, 15531777, 43600150, 122392503, 343575075, 964469468, 2707418035, 7600149781, 21334820094, 59890207635, 168121266303, 471942931900, 1324818304479, 3718974098873

This can also be defined as the expansion of 1/(x^4*p(1/x))) with p = 2-2*x + x^2 - 3*x^3 + x^4.

Limit n -> infinity a(n+1)/a(n) approaches the Pisot root 2.8071578467023438071578467023431323785220673259635911...

G.f.: 1/(1 - 3*x + x^2 - 2*x^3 + 2*x^4).

a(n) = +3*a(n-1) -a(n-2) +2*a(n-3) -2*a(n-4).

(* First method: polynomial expansion*)

Clear[p, q, a] (* one for zero , zero for one matrix substitution in 4x4 theta Pisot matrix :

m0 = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 1}}

m = ((m0 /. 0 -> a) /. 1 -> 0) /. a -> 1*)

m = {{1, 0, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 0}, {0, 1, 1, 0}}

p[x] = Factor[CharacteristicPolynomial[m, x]];

q[x_] = ExpandAll[x^4*p[1/x]];

a = Table[SeriesCoefficient[Series[1/q[x], {x, 0, 50}], m], {m, 0, 50}]

Table[N[a[[n + 1]]/a[[n]]], {n, 1, 50}];

(* second method: vector matrix Markov*)

Clear[m, v, n]

m = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-2, 2, -1, 3}};

vLinearRecurrence[0] = {3, -1, 2, -2}, {1, 3, 8, 23}; , 40] (* _Bruno Berselli_, May 17 2017 *)

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 (3,-1,2,-2).

<a href="/index/Rea#recLCCRec">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-1,2,-2).