On a conjecture of J. Shallit about Ap\'ery-like numbers

TL;DR

This paper confirms a conjecture by J. Shallit regarding the 3-adic valuation of Apéry-like numbers and demonstrates that the resulting sequence is 3-regular, contributing to the understanding of their arithmetic properties.

Contribution

It proves a conjectured formula for the 3-adic valuation of a specific Apéry-like sequence and establishes its 3-regularity.

Findings

Confirmed Shallit's conjecture on the 3-adic valuation formula.

Proved the 3-regularity of the valuation sequence.

Enhanced understanding of the arithmetic structure of Apéry-like numbers.

Abstract

Put $a(n)=\sum\limits_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}$, and $b(n)=v_{3}(a(n))$, for all integers $n\geqslant 0$, where $v_{3}$ is the $3$-adic valuation. In this work, we shall confirm a formula about $b(n)$, conjectured by J. Shallit in 2000. As application, we show that the sequence $(b(n))_{n\geqslant 0}$ is $3$-regular.