Frobenius Numbers and Automatic Sequences

TL;DR

This paper explores the Frobenius numbers associated with certain automatic sequences, using automata theory and logic to compute these numbers for evil, odious, and Wythoff sequences.

Contribution

It introduces a novel automata-theoretic approach to compute Frobenius numbers for classical automatic sequences, contrasting with traditional methods.

Findings

Computed Frobenius numbers for evil, odious, and Wythoff sequences.

Demonstrated automata theory and logic as effective tools for these computations.

Provided new insights into the structure of Frobenius numbers in automatic sequences.

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.