Example 3.10 Let $a_{n}=\left(\frac{5+\sqrt{13}}{2}\right)^{n}-\left(\frac{5-\sqrt{13}}{2}\right)^{n} \quad(n=0,1,2, \cdots)$, try to find a recurrence relation for $a_{n}$.
Solution: Since
$$
\begin{array}{l}
\frac{5+\sqrt{13}}{2}+\frac{5-\sqrt{13}}{2}=5, \\
\frac{5+\sqrt{13}}{2} \cdot \frac{5-\sqrt{13}}{2}=3,
\end{array}
$$

thus $x_{1}=\frac{5+\sqrt{13}}{2}, x_{2}=\frac{5-\sqrt{13}}{2}$ are the two roots of the equation $x^{2}-5 x+3=0$, hence $u_{n}=a_{n}(n=0,1,2, \cdots)$ is a solution to the recurrence relation
$$
u_{n}=5 u_{n-1}-3 u_{n-2} \quad(n \geqslant 2)
$$

Therefore,
$$
a_{n}=5 a_{n-1}-3 a_{n-2} \quad(n \geqslant 2) .
$$