2. In the sequence $\left\{a_{n}\right\}$, it is known that $a_{n+2}=3 a_{n+1}-2 a_{n}, a_{1}=1, a_{2}=3$, then the general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=$
$$
\begin{array}{l}
a_{n+2}-a_{n+1}=2\left(a_{n+1}-a_{n}\right) \Rightarrow a_{n+1}-a_{n}=2^{n} \Rightarrow a_{n}-a_{1}=2+2^{2}+\cdots+2^{n-1} \\
=2^{n}-2 \Rightarrow a_{n}=2^{n}-1\left(n \in \mathbf{N}_{+}\right)
\end{array}
$$

The translation is as follows:

$$
\begin{array}{l}
a_{n+2}-a_{n+1}=2\left(a_{n+1}-a_{n}\right) \Rightarrow a_{n+1}-a_{n}=2^{n} \Rightarrow a_{n}-a_{1}=2+2^{2}+\cdots+2^{n-1} \\
=2^{n}-2 \Rightarrow a_{n}=2^{n}-1\left(n \in \mathbf{N}_{+}\right)
\end{array}
$$