2006 OIM Problems/Problem 6

Problem

Let $n > 1$ be an odd integer. Let $P_0$ and $P_1$ be two consecutive vertices of a regular polygon with $n$ sides. For each $k \ge 2$ , define $P_k$ as the vertex of the given polygon which is located in the bisector of $P_{k−1}$ (Error compiling LaTeX. Unknown error_msg) and $P_{k−2}$ (Error compiling LaTeX. Unknown error_msg). Find for what values of $n$ the sequence $P_0, P_1, P_2,\cdots$ , runs through all the vertices of the polygon.