Narayana sequence and the Brocard–Ramanujan equation

Abstract

Let $\left\lbrace a_{n}\right\rbrace_{n\geq 0}$ be the Narayana sequence defined by the recurrence $a_{n}=a_{n-1}+a_{n-3}$ for all $n\geq 3$ with intital values $a_{0}=0$ and $a_{1}=a_{2}=1$. In this paper, we fully characterize the $3$-adic valuation of $a_{n}+1$ and $a_{n}-1$ and then we find all positive integer solutions $(u,m)$ to the Brocard--Ramanujan equation $m!+1=u^2$ where $u$ is a Narayana number.