Question 78
Let $S=\{(x, y) \in \mathbb{Z}^2 \mid 0 \leq x \leq 11, 0 \leq y \leq 9\}$. Compute the number of sequences $(s_0, s_1, \ldots, s_n)$ of elements in $S$ (for any positive integer $n \geq 2$) that satisfy the following conditions:
- $s_0 = (0,0)$ and $s_1 = (1,0)$,
- $s_0, s_1, \ldots, s_n$ are distinct,
- for all integers $2 \leq i \leq n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^\circ$ or $180^\circ$ in the clockwise direction.
- $s_0 = (0,0)$ and $s_1 = (1,0)$,
- $s_0, s_1, \ldots, s_n$ are distinct,
- for all integers $2 \leq i \leq n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^\circ$ or $180^\circ$ in the clockwise direction.