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MATHEMATICA
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M = {{0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0}, {1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0}, {1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1}, {0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1}, {0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0}};
v[1]= Table[Fibonacci[n], {n, 0, 13}]; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 50}]
LinearRecurrence[{2, 30, -6, -263, -250, 419, 666, 228, -28, -17}, {0, 27, 838, 4025, 29742, 161630, 962784, 5335471, 30120946, 166834881}, 50] (* G. C. Greubel, Jul 22 2023 *)
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