Proving properties of some greedily-defined integer recurrences via automata theory
Discrete Mathematics
2023-08-15 v1 Formal Languages and Automata Theory
Combinatorics
Abstract
Venkatachala on the one hand, and Avdispahi\'c & 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.
Cite
@article{arxiv.2308.06544,
title = {Proving properties of some greedily-defined integer recurrences via automata theory},
author = {Jeffrey Shallit},
journal= {arXiv preprint arXiv:2308.06544},
year = {2023}
}