HMMT November 2017 T5

Ashwin the frog is traveling on the $x y$-plane in a series of $2^{2017}-1$ steps, starting at the origin. At the $n^{\text {th }}$ step, if $n$ is odd, then Ashwin jumps one unit to the right. If $n$ is even, then Ashwin jumps $m$ units up, where $m$ is the greatest integer such that $2^{m}$ divides $n$. If Ashwin begins at the origin, what is the area of the polygon bounded by Ashwin's path, the line $x=2^{2016}$, and the $x$-axis?