Example 6. In the sequence $\left\{a_{n}\right\}$, for any natural number $n(n \geqslant 2)$, we have $a_{n}=3 a_{n-1}-2 a_{n-2}$, and $a_{0}=2, a_{1}=3$, find the general term formula of this sequence.
Solve for $\begin{array}{l} a_{0}=2=2^{0}+1, \\ a_{1}=3=2^{1}+1, \\ a_{2}=3 a_{1}-2 a_{0}=2^{2}+1, \\ a_{3}=3 a_{2}-2 a_{1}=9=2^{3}+1, \cdots, \\ \text { and } 2^{n}+1=(2+1) a_{n-1}-(2 \cdot 1) a_{n-2} \\ =3 a_{n-1}-2 a_{n-2}(n \geqslant 2),\end{array}$