  We define an $n$-gon to be any convex polygon with $n$ vertices. Let $V$
represent the set of vertices of the polygon. A proper $k$-coloring refers to a
function, $f$ : $V$ $\rightarrow$ $\{1, 2, 3, ... k\}$, such that for any two
vertices $u$ and $v$, if $f(u) = f(v)$, $u$ is not adjacent to $v$. The purpose
of this paper is to develop a recursive algorithm to compute the number of
proper $k$-colorings in an $n$-gon. The proposed algorithm can easily be solved
to obtain the explicit expression. This matches the explicit expression
obtained from the popular conventional solutions, which confirms the
correctness of the proposed algorithm. Often, for huge values of $n$ and $k$,
it becomes impractical to display the output numbers, which would consist of
thousands of digits. We report the answer modulo a certain number. In such
situations, the proposed algorithm is observed to run slightly faster than the
conventional recursive algorithm.
