Slow Recurrences

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 $\alpha ,\beta$.