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Rowland, Eric and Yassawi, Reem
(2015).
DOI: https://doi.org/10.5802/jtnb.901
Abstract
In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of Denef and Lipshitz, for computing a finite automaton for the sequence modulo pα, for all but finitely many primes p. This method gives completely automatic proofs of known results, establishes a number of new theorems for well-known sequences, and allows us to resolve some conjectures regarding the Apéry numbers. We also give a second method, which applies to an algebraic sequence modulo pα for all primes p, but is significantly slower. Finally, we show that a broad range of multidimensional sequences possess Lucas products modulo p.