Narayana Sequence and The Brocard-Ramanujan Equation

Abstract

Let ${\left\{a_n\right\}}_{n\ge 0}$ be the Narayana Sequence defined by the recurence $a_n=a_{n-1}+a_{n-3}$ for all $n\ge 3$ 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 prove that there are no integer solutions $(u,m)$ to the Brocard-Ramanujan Equation $m!+1=u^2$ where $u$ is a Narayana number.