3. 2.9 * Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=a_{2}=1, a_{3}=2,3 a_{n+3}=4 a_{n+2}+a_{n+1}-2 a_{n}$, $n=1,2, \cdots$. Find the general term of the sequence $\left\{a_{n}\right\}$.
The characteristic equation is $3 x^{3}-4 x^{2}-x+2=0$, with characteristic roots $x_{1}=1, x_{2}=1, x_{3}=$ $-\frac{2}{3}$, so $a_{n}=\left(A_{1}+A_{2} n\right) x_{1}^{n}+A_{3} x_{3}^{n}$. From the initial conditions, we have $\left\{\begin{array}{l}A_{1}+A_{2}-\frac{2}{3} A_{3}=1, \\ \Lambda_{1}+2 \Lambda_{2}+\frac{4}{9} \Lambda_{3}=1, \\ A_{1}+3 A_{2}-\frac{8}{27} A_{3}=2 .\end{array}\right.$
Solving these, we get $A_{1}=\frac{1}{25}, A_{2}=\frac{3}{5}, A_{3}=-\frac{27}{50}$. Therefore,
$$
a_{n}=\frac{1}{25}\left[1+15 n-\frac{27}{2}\left(-\frac{2}{3}\right)^{n}\right] .
$$