# Motzkin numbers

In previous posts we have established the asymptotic densities of the Motzkin numbers modulo $2^{k}, k \in \{1, 2, 3 \}$ and $5$. The main result of this note is that

Theorem 1.

$\lim_{N \to \infty} \frac {1}{N} \#\{n \leq N: M_n \equiv 0 \mod 3 \} = 1$.

Firstly, let $T(\, 01 )\,$ denote the set of natural numbers which have a base 3 expansion containing only the digits $0$ and $1$. The following theorem from [1] will be used to prove theorem 1.

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

$M_n \equiv -1 \mod 3 \quad$ if  $\quad n \in 3T (\, 01 )\, - 1$,

$M_{n} \equiv 1 \mod 3 \quad$  if  $\quad n \in 3T(\, 01 )\quad$ or $\quad n \in 3T(\, 01 )\, - 2$,

$M_n \equiv 0 \mod 3 \quad$ otherwise.

We will first examine the nature of the set $T(\, 01 \,)$. We have

Theorem 3. The asymptotic density of the set $T(\, 01 \,)$ is zero.

Proof:

Choose $k \in N: 3^{k} \leq N < 3^{k+1}$. Then $k = \lfloor log_3 (\, N )\, \rfloor$ and

$\frac {1}{N} \# \{ n \leq N : n \in T(01) \} \leq \frac {2^{k+1}}{N} \leq \frac {2^{k+1}}{3^{k}} \to 0$ as $N \to \infty$.

$\Box \,$ .

Proof of Theorem 1.

Since the asymptotic density of $T(\, 01 \,)$ is zero, so is the asymptotic density of the sets $3T(\, 01 \,) - k$ for $k \in \{0, 1, 2\}$. Therefore Theorem 2 implies that

$\lim_{N \to \infty} \frac {1}{N} \# \{n \leq N : M_n \equiv +-1 \mod 3 \} = 0$

and the result follows.

$\Box \,$ .

## 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

The Motzkin numbers $M_n$ are defined by

$M_n := \sum_{k \geq 0} \binom {n}{2k} C_{k}$

where $C_{k}\,$ are the Catalan numbers. The following result is established in [1]

Theorem 1 (Theorem 5.5 of [1]). The nth Motzkin number $M_n$ is even if and only if $n = (4i + \epsilon)4^{j+1} - \delta$ for $i, j \in \mathbb{N}, \epsilon \in \{1, 3\}$ and $\delta \in \{1, 2\}$. Moreover, we have

$M_n \equiv 4 \mod 8$ if $(\, \epsilon, \delta )\, = (\, 1, 1 )\,$ or $(\, 3, 2 )\,$

$M_n \equiv 4y + 2 \mod 8$ if $(\, \epsilon, \delta )\, = (\, 1, 2 )\, or (\, 3, 1 )\,$

where $\, y \,$ is the number of digit 1’s in the base 2 representation of $\, 4i + \epsilon - 1$.

Remark 2. The 4 choices for $(\, \epsilon, \delta )\,$ in the above theorem give 4 disjoint sets of numbers $n = (4i + \epsilon)4^{j+1} - \delta$.

Theorem 3. Each of the 4 disjoint sets defined by the choice of $(\, \epsilon, \delta )\,$ in Theorem 1 has asymptotic density $\frac {1}{12}$ in the natural numbers.

Proof: Use the result for the set $S_{1}$ of Theorem 1 of the post here with $q = 4, r = \epsilon, s = 1, t = 1$.

$\Box \,$.

Corollary 4: The asymptotic density of

$\{ n < N: M_n \equiv 0 \mod 2 \}$ is $\frac {1}{3}$.

The asymptotic density of

$\{ n < N: M_n \equiv 4 \mod 8 \}$ is $\frac {1}{6}$.

The asymptotic density of each the sets

$\{ n < N: M_n \equiv 2 \mod 8 \}$ and $\{ n < N: M_n \equiv 6 \mod 8 \}$ is $\frac {1}{12}$.

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 $i$ 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 $4i + \epsilon - 1$. Since the asymptotic density of the 2 sets combined is $\frac {1}{6}$ (from Theorems 1 and 3), each of the two sets has asymptotic density of $\frac {1}{12}$.

## 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

The Motzkin numbers $M_n$ are defined by

$M_n := \sum_{k \geq 0} \binom {n}{2k} C_{k}$

where $C_{k}\,$ are the Catalan numbers. The following result is established in [1]

Theorem 1 (Theorem 5.4 of [1]). The Motzkin number $M_n$ is divisible by $5$ if and only if $n$ is one of the following forms

$(\, 5i + 1 )\, 5^{2j} - 2,\, (\, 5i + 2 )\, 5^{2j-1} - 1,\, (\, 5i + 3 )\, 5^{2j-1} - 2,\, (\, 5i + 4 )\, 5^{2j} - 1$

where $i, j \in \mathbb{N}$ and $j \geq 1$.

Theorem 2: The asymptotic density of numbers of the first form in theorem 1 is $\frac {1}{120}$. Numbers of the fourth form also have asymptotic density $\frac {1}{120}$. The asymptotic density of numbers of the second and third forms in theorem 1 is $\frac {1}{24}$ each.

Corollary 3: The asymptotic density of $\#\{n < N: M_{n} \equiv 0 \mod {5} \}$ is $\frac {1}{10} \,$.

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 $1, 2, 3, 4 \mod {5}$.

Proof of Theorem 2.

Firstly consider numbers of the form $(\, 5i + 1)\, 5^{2j} - 2\,$. As we are interested in asymptotic density it is enough to look at numbers of the form $(\, 5i + 1)\, 5^{2j}\,$. We have, for fixed $j \geq 1$,

$\#\{ i \geq 0: (\, 5i + 1)\, 5^{2j} \leq N \} = \frac {N}{5^{2j+1}} - \frac {1}{5} - E(\, j, N )\,$

where $0 \leq E( \, j, N ) < 1$ is an error term introduced by not rounding down to the nearest integer. So,

$\#\{n < N: n = (\, 5i + 1)\, 5^{2j} \, for some \, i, j \in \mathbb{N} \, with \, j \geq 1 \}$

$= \sum_{j \geq 1} (\, \frac {N}{5^{2j+1}} - \frac {1}{5} - E(\, j, N )\, )\, = \sum_{j = 1}^{\lfloor \frac{1}{2} (\, \log_{5} (\, N )\,- 1 )\, \rfloor} (\, \frac {N}{5^{2j+1}} - \frac {1}{5} - E(\, j, N )\, )\,$

$= \frac {N}{125} \sum_{j = 0}^{\lfloor \frac{1}{2} (\, \log_{5} (\, N )\,- 1 )\, \rfloor - 1} (\, \frac {1}{25} )\,^{j} - E^{'}(\, N )\,$

where the new error term $E^{'} (\, N )\,$ satisfies $0 \leq E^{'} (\, N )\, < \frac{3}{5} (\, \log_{5} (\, N )\,$. Then

$\#\{n < N: n = (\, 5i + 1)\, 5^{2j} \, for some \, i, j \in \mathbb{N} \, with \, j \geq 1 \}$

$= (\, \frac {N}{125} )\, \frac {1 - (\, \frac {1}{25} )\,^{\lfloor \frac{1}{2} (\, \log_{5} (\, N )\,- 1 )\, \rfloor} }{1 - \frac {1}{25} } - E^{'}(\, N )\,$.

Since $\lim_{N \to \infty} \frac {E^{'}(\, N )\, }{N} = 0$

$\lim_{N \to \infty} \frac {1}{N} \#\{n < N: n = (\, 5i + 1)\, 5^{2j} \, for some \, i, j \in \mathbb{N} \, with \, j \geq 1 \} = \frac {1}{120}$.

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

$\Box$.

## 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