2. Given the sequence $\left\{a_{n}\right\}$, the first 4 terms are $a_{1}=2, a_{2}=3, a_{3}=5, a_{4}=8$. Please choose one of the following ways to answer:
(1) Write a recursive formula for the sequence $\qquad$ ;
(2) Write a general term formula for the sequence $\qquad$ .
2. (1) $a_{n+2}=a_{n+1}+a_{n}, a_{n+1}-a_{n}=n$ etc.
$$
\begin{array}{l}
\text { (2) } a_{n}=\frac{n^{2}-n+4}{2} \\
a_{n}=\frac{n^{2}-n+4}{2}+(n-1)(n-2)(n-3)(n-4) b_{n} \\
a_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+2}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+2}\right] .
\end{array}
$$