Asymptotic density of Motzkin numbers modulo 3

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

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: