# Asymptotic densities of Motzkin numbers modulo 5

The Motzkin numbers are defined by

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

**Theorem 1 (Theorem 5.4 of [1]). **The Motzkin number is divisible by if and only if is one of the following forms

where and .

**Theorem 2: **The asymptotic density of numbers of the first form in theorem 1 is . Numbers of the fourth form also have asymptotic density . The asymptotic density of numbers of the second and third forms in theorem 1 is each.

**Corollary 3: **The asymptotic density of is .

**Remark 4**: Corollary 3 follows immediately from theorems 1 and 2 and the disjointness of the 4 forms of integers listed in theorem 1.

**Remark 5**: Numerical tests also show that roughly 22.5% of Motzkin numbers are congruent to each of .

**Proof of Theorem 2.**

Firstly consider numbers of the form . As we are interested in asymptotic density it is enough to look at numbers of the form . We have, for fixed ,

where is an error term introduced by not rounding down to the nearest integer. So,

where the new error term satisfies . Then

.

Since

.

A similar argument can be used to establish the asymptotic densities of the other 3 forms of numbers in the theorem.

**.**

## 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