= Solution
For the rotation group $C_5$, the identity fixes all $n^5$ colorings. Because five is prime, each of the four nonidentity rotations is a single five-cycle and fixes exactly the $n$ constant colorings. <Burnside lemma> therefore gives the number of <coloring necklaces>
$$
\boxed{\frac{n^5+4n}{5}}.
$$
For the <dihedral group> $D_{10}$, the five reflections each fix one vertex and interchange two pairs of vertices, so each fixes $n^3$ colorings. Adding these to the identity and rotation contributions gives the number of <coloring bracelets>
$$
\boxed{\frac{n^5+5n^3+4n}{10}}.
$$
Back to article page
