Slow Recurrences

Sam Spiro

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arXiv · 1909.06517

Slow Recurrences

Sam Spiro

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Abstract

For positive integers $α$ and $β$, we define an $(α,β)$-walk to be any sequence of positive integers satisfying $w_{k+2}=αw_{k+1}+βw_k$. We say that an $(α,β)$-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 $(α,β)$-walks for arbitrary positive $α,β$. 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 $α,β$.

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BibTeX RIS

Sam Spiro. 2019-09-14. Slow Recurrences. https://arxiv.org/abs/1909.06517

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