Question 5
For a positive integer $n \ge 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move. Prove that $a_{n-1} + a_n = 2^n$ for all $n \ge 4$.
(This problem was suggested by Sam Vandervelde.)
(This problem was suggested by Sam Vandervelde.)