Putnam 1993 A6

The infinite sequence of 2's and 3's

$$\begin{aligned}&2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ &3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\dots\end{aligned}$$

has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1 + \lfloor rm \rfloor$ for some nonnegative integer $m$. (Note: $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$.)