More precisely, this is the number of ways of making a list of the 2^n nodes of the n-cube, with a distinguished starting position and a direction, such that each node is adjacent to the previous one. The final node may or may not be adjacent to the first. Finally, divide by 2^n since the starting node really doesn't matter.
Also, the number of strings s of length 2^n - 1 over the alphabet {1,2,...,n} with the property that every contiguous subblock has some letter that appears an odd number of times.
