Non equivalent colorings for Isosceles Triangle

Question

Using Burnside algorithm the number of non equivalent colorings using k colors for an equilateral triangle is given as $\frac{k^3+3k^2+2k}6$. Is there any formula that has been derived for an Isosceles triangle where symmetry is less compared to an Equilateral triangle.

Answer 1

A (non-equilateral) isosceles triangle has only two symmetries, the identity and a reflection. These fix $k^3$ and $k^2$ colourings respectively, so the number of non-equivalent colourings is $\frac{k^3+k^2}2$.