2013 USAMO 第 2 题

For a positive integer $n\geq 3$ 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 $n\geq 4$.

对于正整数 $n\geq 3$ 围绕圆绘制 $n$ 个等距点。将其中一个标记为 $A$，并在 $A$ 处放置一个标记。人们可以将标记沿顺时针方向向前移动到下一个点或下一个点。因此总共有 $2n$ 可用的不同动作；每个点有两个。让 $a_n$ 计算在 $A$ 处开始和结束且不重复移动的情况下绕圆圈前进两次的方式数量。证明 $a_{n-1}+a_n=2^n$ 对于所有 $n\geq 4$。