Question 137
Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n\gt 0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \times 3$ grid, and then removing the center copy. For example, $S_{3}$ is shown below:
Let $a_{n}$ be the expected value of $|x-x'|+|y-y'|$, where $(x, y)$ and $(x', y')$ are two points chosen randomly within $S_{n}$. There exist relatively prime positive integers $a$ and $b$ such that
$$
\lim_{n \rightarrow \infty} \frac{a_{n}}{3^{n}}=\frac{a}{b} .
$$
Compute $100 a+b$.
Let $a_{n}$ be the expected value of $|x-x'|+|y-y'|$, where $(x, y)$ and $(x', y')$ are two points chosen randomly within $S_{n}$. There exist relatively prime positive integers $a$ and $b$ such that
$$
\lim_{n \rightarrow \infty} \frac{a_{n}}{3^{n}}=\frac{a}{b} .
$$
Compute $100 a+b$.