The recent posts regarding Catalan and Motzkin numbers have been consolidated in two papers at arXiv. They are located at https://arxiv.org/pdf/1611.04910v1.pdf (Asymptotic density of Motzkin numbers modulo small primes) and https://arxiv.org/pdf/1611.03705v1.pdf (Asymptotic density of Catalan numbers modulo 3 and powers of 2).
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
This note will establish formulae for the asymptotic densities in the natural numbers of two sets of numbers. These sets are
for integers .
Theorem 1. Let with and . Then the asymptotic density of the set is . In addition the asymptotic density of the set is .
We firstly look at . We have, for fixed ,
where is an error term introduced by not rounding down to the nearest integer. So, letting
where the new error term satisfies .
The proof for is very similar. With and defined as above we have for fixed ,
The Motzkin numbers are defined by
where are the Catalan numbers. The following result is established in 
Theorem 1 (Theorem 5.5 of ). The nth Motzkin number is even if and only if for and . Moreover, we have
where is the number of digit 1’s in the base 2 representation of .
Remark 2. The 4 choices for in the above theorem give 4 disjoint sets of numbers .
Theorem 3. Each of the 4 disjoint sets defined by the choice of in Theorem 1 has asymptotic density in the natural numbers.
Proof: Use the result for the set of Theorem 1 of the post here with .
Corollary 4: The asymptotic density of
The asymptotic density of
The asymptotic density of each the sets
and is .
Remark 5: The first 2 statements of Corollary 4 follows immediately from theorems 1 and 3. The third statement follows from the observation that the numbers of 1’s in the base 2 expansion of is equally likely to be odd or even and therefore the same applies the the number of 1’s in the base 2 expansion of . Since the asymptotic density of the 2 sets combined is (from Theorems 1 and 3), each of the two sets has asymptotic density of .
 Eu, S.-P., Liu, S.-C., & Yeh, Y.-N. (2008). Catalan and Motzkin numbers modulo 4 and 8. European Journal of Combinatorics, 29(6), 1449–1466. http://doi.org/10.1016/j.ejc.2007.06.019