Given two relatively prime integers a and b, is there an easy characterization for when a^2+b^2 is square free?

Edit: The above question proved to be too general. The problem I had in my mind is as follows: given the two sequences $a_n$ and $b_n$ defined by $a_0=b_0=1$, $a_{n+1}=a_nb_n,\ b_{n+1}=a_n^2+b_n^2$, are the $b_n$'s square free and coprime?