HMMT 2013 U21

Find the number of positive integers $j \leq 3^{2013}$ such that

$$j=\sum_{k=0}^{m}\left((-1)^{k} \cdot 3^{a_{k}}\right)$$

for some strictly increasing sequence of nonnegative integers $\left\{a_{k}\right\}$. For example, we may write $3=3^{1}$ and $55=3^{0}-3^{3}+3^{4}$, but 4 cannot be written in this form.