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arXiv · 2103.10904

Frobenius Numbers and Automatic Sequences

Jeffrey Shallit
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Abstract

The Frobenius number $g(S)$ of a set $S$ of non-negative integers with $\gcd 1$ is the largest integer not expressible as a linear combination of elements of $S$. Given a sequence ${\bf s} = (s_i)_{i \geq 0}$, we can define the associated sequence $G_{\bf s} (i) = g(\{ s_i,s_{i+1},\ldots \})$. In this paper we compute $G_{\bf s} (i)$ for some classical automatic sequences: the evil numbers, the odious numbers, and the lower and upper Wythoff sequences. In contrast with the usual methods, our proofs are based largely on automata theory and logic.

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Jeffrey Shallit. 2021-03-19. Frobenius Numbers and Automatic Sequences. https://arxiv.org/abs/2103.10904

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