Consider a Double Tower of Hanoi. In this variation of the Tower of Hanoi there are three poles in a row and 2n disks, two of each of n different sizes, where n is any positive integer. Assume one of the poles initially contains all of the disks placed on top of each other in pairs of decreasing size. Disks are transferred one by one from one pole to another, but at no time may a larger disk be placed on top of a smaller disk. However, a disk may be placed on top of one of the same size. Let an be the minimum number of moves needed to transfer a tower of 2n disks from one pole to another. (a) Find a1, a2, and a3.
(b) Derive a recurrence relation for the sequence a1, a2, and a3 and check that the results in (a) satisfy your recurrence relation.
(c) Show that the formula an = 2n+1 - 2 for the nth term in the sequence satisfies the recurrence relation you derived in (b).