Consider $n$ equally spaced points along a circle. Paint each point with any of $m$ distinct colors. There are $m^n$ ways of doing so, if we consider each assignment of colors to points as distinct. But suppose we identify colorings that are related by a rotation and/or reflection. Consider two color assignments to be equivalent if they are related by a reflection and/or rotation, and inequivalent otherwise. How many inequivalent color assignments are there?
I imagine this will involve some combinatorics and group theory, assuming a clean expression involving $n$ and $m$ exists to begin with. This question was inspired by a fidget spinner toy I was given recently, where each of three prongs contains an LED that can be set to any of three modes, or be left off, giving four distinct possibilities per prong. Someone asked me how many "distinct" settings exist. It was easy enough to count them all in a case-by-case approach: There are 20, unless I made a mistake. But it got me wondering if there are combinatorics formulas that could describe the generalized case.
