On the matrix powers and exponential by the r-generalized Fibonacci sequences methods: the companion matrix case

AbstractProperties of some r-generalized Fibonacci sequences in the algebra of square matrices GL(r,C), allow us to derive some explicit formulas for the matrix powers An (n⩾0) and exponential etA, for every A∈GL(r,C). Connection with the Verde-Star method is displayed. For a companion matrix A, we present some explicit combinatorial formulas for An (n⩾0) and etA. Furthermore, the Chen–Louck’s Theorem is derived