In previous posts we have established the asymptotic densities of the Motzkin numbers modulo and . The main result of this note is that

**Theorem 1. **

.

Firstly, let denote the set of natural numbers which have a base 3 expansion containing only the digits and . The following theorem from [1] will be used to prove theorem 1.

**Theorem 2.** (Corollary 4.10 of [1]). The Motzkin numbers satisfy

if ,

if or ,

otherwise.

We will first examine the nature of the set . We have

**Theorem 3.** The asymptotic density of the set is zero.

**Proof: **

Choose . Then and

as .

** .**

**Proof of Theorem 1.**

Since the asymptotic density of is zero, so is the asymptotic density of the sets for . Therefore Theorem 2 implies that

and the result follows.

** .**

## References.

[1] Deutsch, E., & Sagan, B. E. (2006). Congruences for Catalan and Motzkin numbers and related sequences. Journal of Number Theory, 117(1), 191–215. http://doi.org/10.1016/j.jnt.2005.06.005