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 $\alpha,\beta$.

中文翻译：

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缓慢复发

对于正整数 $\alpha$ 和 $\beta$，我们将 $(\alpha,\beta)$-walk 定义为满足 $w_{k+2}=\alpha w_{k+1 的任何正整数序列}+\beta w_k$。如果 $w_s=n$ 且 $s$ 尽可能大，我们说 $(\alpha,\beta)$-walk 是 $n$-slow。一些作者已经研究了慢 $(1,1)$-walks。在本文中，我们考虑对任意正值 $\alpha,\beta$ 的 $(\alpha,\beta)$-walks。我们推导出这些游走的特征定理，并由此证明了关于给定 $n$ 的 $n$-slow 游走总数的几个结果。除此之外，我们研究了所有可能的 $\alpha,\beta$ 中给定的 $n$ 中最慢的 $n$-slow walk。