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).

# Uncategorized

# Asymptotic density of Motzkin numbers modulo 8

The Motzkin numbers are defined by

where are the Catalan numbers. The following result is established in [1]

**Theorem 1** (Theorem 5.5 of [1]). The nth Motzkin number is even if and only if for and . Moreover, we have

if or

if

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

is .

The asymptotic density of

is .

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 .

## Reference

[1] 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