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A representation and some properties for \(k\)-Fibonacci sequences. (English) Zbl 1028.11011

Summary: The \(k\)-Fibonacci sequence \(\{g_n^{(k)}\}\) is defined as: \[ g_1^{(k)}=\dots= g_{k-2}^{(k)}= 0, \quad g_{k-1}^{(k)}= g_k^{(k)}= 1 \] and for \(n> k\geq 2\), \[ g_n^{(k)}= g_{n-1}^{(k)}+ g_{n-2}^{(k)} +\cdots+ g_{n-k}^{(k)}. \] In this paper, we give a combinatorial representation of \(g_n^{(k)}\) and give some properties for \(k\)-Fibonacci sequences.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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