Adapted from a past Putnam problem

Let $(a_n)_{n=1}^{\infty}$ be a sequence of $1$ s and $2$ s such that $a_1 = 1$ and $a_n$ is the number of $2s$ between the $n^{th}$ and $(n+1)^{th}$ $1s$ in the sequence. So the sequence starts out like this: $1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, ...$

Prove (if you wish) that there exists a real number $r$ such that $a_n = 1$ if and only if $n = 1 + \lfloor rk \rfloor$ for some $k \in \mathbb{N}$ .

If $r$ can be written in simplest form as $r = \frac{\alpha+\sqrt{\beta}}{\gamma}$ where $\alpha, \beta, \gamma$ are coprime natural numbers, find $\alpha^2+\beta^2+\gamma^2$ .