IT-CESENATICO 2005 Problem 5

Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and

$$a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }a_n+h \text{ otherwise }.\end{array}$$

For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$ ?