Find the chromatic polynomial of $P_3\times P_n$

Question

I understand the concept of the chromatic polynomial of the ladder graph $P_2\times P_n$ but I can not find the chromatic polynomial of $P_3\times P_n$.

Answer 1

I doubt there is any nice formula. The first few are: \begin{matrix} n & \text{chromatic polynomial of $P_3 \times P_n$}\\ \hline 1 &x (x-1)^2 \\ 2 &x (x - 1)(x^2 - 3 x + 3)^2 \\ 3 &x (x - 1) (x^7 - 11 x^6 + 55 x^5 - 161 x^4 + 298 x^3 - 350 x^2 + 244 x - 79) \end{matrix} The first 30 are here.