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](http://oeis.org/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 $n \geq n\_0$, $a(n) = a(n\_0 + (n - n\_0)\bmod k))$ ?



Collatz dropping times aperiodicity
$a(n)=\lfloor \alpha n\rfloor-\lfloor \alpha(n-1)\rfloor$, where $\alpha=\log\_23$. Sequences of this type are called Sturmian and are well known (and easily shown) not to be periodic. Basically: for any $x,x'$ whose difference is not an integer, there is a multiple of $\alpha$ such that $\lfloor x+n\alpha\rfloor-\lfloor x\rfloor\ne
\lfloor x'+n\alpha\rfloor-\lfloor x'\rfloor$ by density of multiples of $\alpha$ in the circle. 


