In previous posts we have established the asymptotic densities of the Motzkin numbers modulo and . The main result of this note is that
Firstly, let denote the set of natural numbers which have a base 3 expansion containing only the digits and . The following theorem from  will be used to prove theorem 1.
Theorem 2. (Corollary 4.10 of ). The Motzkin numbers satisfy
if or ,
We will first examine the nature of the set . We have
Theorem 3. The asymptotic density of the set is zero.
Choose . Then and
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.
 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