Suppose \(d(\boldsymbol{a},\boldsymbol{b})=k\) for some \(k\). Try to construct a path of length \(k\) in the natural graph from the vertex labelled \(\boldsymbol{a}\) to the vertex labelled \(\boldsymbol{b}\).
For fixed \(\boldsymbol{a} \in A_n\), how many \(\boldsymbol{b} \in A_n\) have the property that \(d(\boldsymbol{a},\boldsymbol{b})=k\)?
Find a function that works for \(n=2\) and use this to build one for \(n=3\). It might be useful to think of the natural graph of \(A_2\) as a square and the natural graph of \(A_3\) as a cube. As well, a cube can be thought of as two squares on top of each other with vertical edges connecting corresponding vertices in the top and bottom faces.