How can I prove that the Wiener index of the hypercube graph is $k \cdot 4^{k-1}$? I'm trying to find the Wiener index of the Hypercube graph, $Q_k$. In case anyone does not know what that is, it's a graph where every vertex is labelled by a binary $k$-tuple, and where two vertices are connect only if the tuples differ in only one place. So for example, $Q_2$ would have vertices labelled $(0, 0), (0, 1), (1, 0)$ and $(1, 1)$. $(0, 0)$ would be connected to $(0, 1)$ and $(1, 0)$, and so on.
I wrote out a few examples and I think that the Wiener index of the graph can be represented by $k \cdot  4^{k-1}$. But how do I prove this? The only idea that occurs to me is to use induction, but I am not entirely sure how to go about doing this. Any help with this would be appreciated. If anyone has any other ideas for a proof, that would be fine too.

For any two graphs $G$ and $H$ we have $$ W(G \square H ) = |V(G)|^2 W(H) + |V(H)|^2W(G),$$ where $W(G)$ is the Wiener index of $G.$
Now the hypercube graph $Q_k = Q_{k-1} \square K_2$ and the result now immediately follows by induction.
