Let $a(n)$ for $n \geq 1$ be the number of powers of two $2^m$ that lie between $3^{n-1}$ and $3^{n}$:

$$a(n) = 1, 2, 1, 2, 1, 2, 2, 1, ...$$

It represents the increments between successive terms of allowable dropping times in the Collatz ($3x+1$) problem (see oeis A022921).

Is there an easy way to prove that the sequence never becomes periodic? i.e. do not exist $n_0, k \geq 1$ such that for all $n \geq n_0$, $a(n) = a(n_0 + (n - n_0)\bmod k))$ ?