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MATHEMATICA
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M = {{0, 1, 0, 0, 0, 1, 1}, {1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 1, 0, 0, 1}, {0, 0, 1, 0, 1, 0, 1}, {0, 0, 0, 1, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 0}};
v[1] = {0, 1, 1, 2, 3, 5, 8}; v[n_]:= v[n]= M.v[n-1];
Table[v[n][[1]], {n, 50}]
LinearRecurrence[{0, 11, 12, -10, -12}, {0, 14, 32, 150, 492}, 40] (* G. C. Greubel, Jul 22 2023 *)
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