Structure and asymptotics for Motzkin numbers modulo primes using automata

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.

arXiv comment21 pages, 4 figures, 6 tables. arXiv admin note: text overlap with arXiv:1612.08146, arXiv:1701.02975