Frobenius Numbers and Automatic Sequences

03/19/2021

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by   Jeffrey Shallit, et al.

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The Frobenius number g(S) of a set S of non-negative integers with 1 is the largest integer not expressible as a linear combination of elements of S. Given a sequence s = (s_i)_i ≥ 0, we can define the associated sequence G_ s (i) = g({ s_i,s_i+1,…}). In this paper we compute G_ 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.