Proposition 6 Use $m(m \geqslant 2)$ different colors to color the $n+1(n \geqslant 4)$ interconnected regions $A_{0}$, $A_{1}$, $A_{2}$, $\cdots$, $A_{n}$ in Figure 3, and any two regions separated by 3 other regions, except for $A_{0}$, are colored differently. The number of different coloring schemes is
$a_{n}=\left\{\begin{array}{l}\left.m\left[(m-1)^{\frac{n}{4}}+(-1)^{\frac{n}{4}}(m-1)\right]^{4}, \quad n \equiv 0 \pmod{4}, \\ m\left[(m-1)^{n}+(-1)^{n}(m-1)\right], \quad n \equiv 1 \pmod{4} \text { or } n \equiv 3 \pmod{4}, \\ m\left[(m-1)^{\frac{n}{2}}+(-1)^{\frac{n}{2}}(m-1)\right]^{2}, \quad n \equiv 2 \pmod{4} .\end{array}\right.$