Cyclic sequence is equivalence class of cyclic shift action.

If $a = (a_1, ... , a_i)_c$ is cyclic sequence then  $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, a_1, \ldots , a_{i-2}, a_{i-1})_c$.

Let $i = 2^n$, $a_j \in \{0,1,2,3,4\}$, and $\forall a$ $|\{a_j:a_j \in A \}| < 4$.

I want to find number and asymptotic for all such cyclic sequences.

Thank you for any help!