I see why it stays whole. The $2^-1$ term is offset by the addition of 1 to create a carry. That said, this carry does not exist / work after the second iteration when you substitute, because you are working with 3n+2. I am finding that it can only continue forever given this substitution because in the case of 2n+1 where n = 0, 0 has an infinite number of factors. There are no other whole integers that can claim this.