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MATHEMATICA
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M = {{0, 1, 1, 1, 1, 0}, {1, 0, 1, 0, 1, 1}, {1, 1, 0, 1, 0, 1}, {1, 0, 1, 0, 1, 1}, {1, 1, 0, 1, 0, 1}, {0, 1, 1, 1, 1, 0}} v[1] = {0, 1, 1, 2, 3, 5} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}] Det[M - x*IdentityMatrix[6]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[6]] == 0, x][[n]], {n, 1, 6}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]
LinearRecurrence[{2, 8}, {7, 34}, 30] (* Harvey P. Dale, Sep 06 2016 *)
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