On the $3$-adic valuation of the Narayana numbers

TL;DR

This paper investigates the 3-adic valuation of Narayana numbers, a sequence defined by a specific recurrence, and characterizes which Narayana numbers are factorials.

Contribution

It provides a complete characterization of the 3-adic valuation of Narayana numbers and identifies all Narayana numbers that are factorials.

Findings

Explicit formula for the 3-adic valuation of Narayana numbers

Complete list of Narayana numbers that are factorials

Insights into the divisibility properties of the sequence

Abstract

We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the $3$-adic valuation of ${a_n}$ and use this to determine all Narayana numbers that are factorials.