Counting non-isomorphic maximal independent sets of the n-cycle graph

It is known that the number of maximal independent sets of the $n$-cycle graph $C_n$ is given by the $n$th term of the Perrin sequence. The action of the automorphism group of $C_n$ on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabelled (i.e., defined up to a rotation) maximal independent sets and the number of unlabelled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences.