Abstract
In this paper, using elementary symmetric function and complete homogeneous symmetric function, we obtain a determinant formula for the k-generalized Fibonacci sequence. The relationship between the k-generalized Fibonacci sequence and the order- k linear homogeneous recurrence relation has been investigated, and a general solution for the latter is also derived. Furthermore, we obtain an explicit expression for the elements in the nth power of the companion matrix in terms of k-generalized Fibonacci numbers.
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