3.2.7 * Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=3, a_{2}=8, a_{n+2}=2 a_{n+1}+2 a_{n}, n=1,2, \cdots$. Find the general term $a_{n}$ of the sequence $\left\{a_{n}\right\}$.
The corresponding characteristic equation is $x^{2}=2 x+2$, with characteristic roots $x_{1}=1+\sqrt{3}, x_{2}=1-\sqrt{3}$. Thus, $a_{n}=A_{1}(1+\sqrt{3})^{n}+A_{2}(1-\sqrt{3})^{n}$. From the initial conditions, we have
$$
\left\{\begin{array}{l}
\Lambda_{1}(1+\sqrt{3})+\Lambda_{2}(1-\sqrt{3})=3, \\
A_{1}\left(1+\sqrt{3}\right)^{2}+A_{2}(1-\sqrt{3})^{2}=8,
\end{array}\right.
$$

Solving these, we get $A_{1}=\frac{2+\sqrt{3}}{2 \sqrt{3}}, A_{2}=\frac{\sqrt{3}-2}{2 \sqrt{3}}$. Therefore, $a_{n}=\frac{2+\sqrt{3}}{2 \sqrt{3}}(1+\sqrt{3})^{n}+\frac{\sqrt{3}-2}{2 \sqrt{3}}(1-\sqrt{3})^{n}$.