HMMT 2011 U25

Let $n$ be an integer greater than 3 . Let $R$ be the set of lattice points $(x, y)$ such that $0 \leq x, y \leq n$ and $|x-y| \leq 3$. Let $A_{n}$ be the number of paths from $(0,0)$ to $(n, n)$ that consist only of steps of the form $(x, y) \rightarrow(x, y+1)$ and $(x, y) \rightarrow(x+1, y)$ and are contained entirely within $R$. Find the smallest positive real number that is greater than $\frac{A_{n+1}}{A_{n}}$ for all $n$.