Proof Writing & Problem Solving · Superchallenge problem

A robot starts at the origin facing east. On its $k$-th move ($k = 1, 2, 3, \ldots$) it walks exactly $k$ units straight ahead, and after every move it turns $90^\circ$, left or right as it chooses. For how many $n$ with $1 \le n \le 100$ can the robot be back at the origin immediately after its $n$-th move?