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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
##### Keywords:
Fibonacci; recurrence; $$(\alpha,\beta)$$-walk
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##### References:
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