  We delve into the connection between base $\frac{3}{2}$ and the greedy
partition of non-negative integers into 3-free sequences. Specifically, we find
a fractal structure on strings written with digits 0, 1, and 2. We use this
structure to prove that the even non-negative integers written in base
$\frac{3}{2}$ and then interpreted in base 3 form the Stanley cross-sequence,
where the Stanley cross-sequence comprises the first terms of the infinitely
many sequences that are formed by the greedy partition of non-negative integers
into 3-free sequences.
