It is well-known that there is a formula for the number of necklaces when rotations are identified (but reflections are distinct): $$Z_n = \dfrac{1}{n} \displaystyle \sum \limits_{d \mid n} \phi \left( d \right) 2^{n/d}.$$
I wonder how to count the number when reflections are also considered the same. I believe someone must have asked about this here but I only see thoes without reflections.
Related:
You can use Burnside's lemma for that, as well. The number of fixed points of reflections can be computed easily, there is only one tricky part you should be careful with.
If $n$ is odd, then the axes of all $n$ reflections connect a vertex and the midpoint of the opposite side. So there are $2^\frac{n+1}{2}$ colorings that are fixed by such a reflection, as you can independently choose the color of $\frac{n+1}{2}$ consecutive vertices, and the rest is uniquely determined.
If $n$ is even, then make a case distinction: there are $n/2$ reflections whose axes connect two vertices, and $n/2$ reflections whose axes connect two midpoints.
Can you finish from here?
