In a Triple Tower of Hanoi, There Are Three Poles $t _ { n}$

Essay

In a Triple Tower of Hanoi, there are three poles in a row and 3n disks, three of each of n
different sizes, where n is any positive integer. Initially, one of the poles contains all the disks
placed on top of each other in triples 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 $t _ { n}$
be the minimum number of moves needed to transfer a tower of 3n disks from one pole to another. Find a recurrence relation for $t _ { 1 } , t _ { 2 } , t _ { 3 } , \ldots$ Justify your answer carefully.