If $S=(s_1,s_2,\ldots)$ is defined by $s_ i=\max\{j\in\mathbb{Z}_{\ge0}|2^j\text{ divides }i\}$, then no subsequence of $S$ can appear twice in a row.

Question

Let $S = (s_1, s_2, s_3, \ldots)$ be an infinite sequence where $$s_ i = \max\big\{j \in \mathbb{N} \cup \{0\} \big|2^j\text{ divides }i\big\}\,.$$ I need to prove (i'm quite sure it's true) that a subsequence $S'=x_1, x_2, \ldots, x_k$ of $S$ can't appear twice in a row for any $k$, meaning that $S'S'$ can't appear in the sequence $S$

I thought maybe it has something with the even numbers and odd numbers in the sequence. I tried to prove it using induction, but failed to prove anything in the induction step.I'm also thinking it was a naive try.