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Slow recurrences. (English) Zbl 07257234
Summary: For positive integers \(\alpha\) and \(\beta\), we define an \((\alpha,\beta)\)-walk to be any sequence of positive integers satisfying \(w_{k+2}=\alpha w_{k+1}+\beta w_k\). We say that an \((\alpha,\beta)\)-walk is \(n\)-slow if \(w_s=n\) with \(s\) as large as possible. Slow \((1,1)\)-walks have been investigated by several authors. In this paper we consider \((\alpha, \beta)\)-walks for arbitrary positive \(\alpha, \beta \). We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of \(n\)-slow walks for a given \(n\). In addition to this, we study the slowest \(n\)-slow walk for a given \(n\) amongst all possible \(\alpha,\beta\).
MSC:
11B37 Recurrences
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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