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This paper concerns the Fibonacci-Horner decomposition of the matrix powers A n and the matrix exponential e tA (A ∈ M (r; C), t ∈ R), which is derived from the combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. More precisely, e tA is expressed in a natural way in the so-called Fibonacci-Horner basis with the aid of the dynamical solution of the associated ordinary differential equation. Two simple processes for computing the dynamical solutiondoi:10.13001/1081-3810.1228 fatcat:4e7xekra35cfhjwfcmu7zvmmy4