Proving properties of some greedily-defined integer recurrences via automata theory

08/12/2023

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by   Jeffrey Shallit, et al.

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Venkatachala on the one hand, and Avdispahić Zejnulahi on the other, both studiied integer sequences with an unusual sum property defined in a greedy way, and proved many results about them. However, their proofs were rather lengthy and required numerous cases. In this paper, I provide a different approach, via finite automata, that can prove the same results (and more) in a simple, unified way. Instead of case analysis, we use a decision procedure implemented in the free software Walnut. Using these ideas, we can prove a conjecture of Quet and find connections between Quet's sequence and the "married" functions of Hofstadter.