Structure and asymptotics for Motzkin numbers modulo primes using automata

TL;DR

This paper investigates the divisibility properties of Motzkin numbers modulo primes greater than 3, establishing bounds on their asymptotic density and characterizing cases of divisibility using automata theory.

Contribution

It introduces new bounds and criteria for Motzkin numbers' divisibility by primes, employing automata methods for analysis.

Findings

Lower bound of 2/p(p-1) for divisibility density

Criteria for when the density equals 1

Partial characterization of divisible Motzkin numbers

Abstract

We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic density is actually 1. We also provide a partial characterisation of those Motzkin numbers which are divisible by a prime p > 3. All results are obtained using the automata method of Rowland and Yassawi.