Three, (50 points) Try to find the number of all sequences that satisfy the following conditions: (1) each term is an integer no less than 2;
(2) the sum of all terms equals a fixed value $m$.


Three, Solution: A sequence that satisfies conditions (1) and (2) is called an "$f(m)$" sequence. We now seek the number of such $f(m)$ sequences:

For a sequence $\left\{a_{n}\right\}$ with $k$ terms, there is $a_{1}+a_{2}+\cdots+a_{k}=m\left(a_{i} \geqslant 2\right)$ (1). The number of sequences $\left\{a_{n}\right\}$ is the number of integer solutions to equation (1) that satisfy $a_{i} \geqslant 2(i=1,2, \cdots, k)$, which is also the number of non-negative integer solutions to the following equation: $x_{1}+x_{2}+\cdots+x_{k}=m-2 k(2)$.

Since the number of non-negative integer solutions to equation (2) is $\mathrm{C}_{m-k-1}^{k-1}$, the number of $f(m)$ sequences is: $|f(m)|=$
$$
\begin{array}{l}
\quad \mathrm{C}_{m-2}^{0}+\mathrm{C}_{m-3}^{1}+\mathrm{C}_{m-4}^{2}+\mathrm{C}_{m-5}^{3}+\cdots=\mathrm{C}_{m-2}^{0}+\left(\mathrm{C}_{m-4}^{1}\right. \\
\left.+\mathrm{C}_{m-4}^{0}\right)+\left(\mathrm{C}_{m-5}^{2}+\mathrm{C}_{m-5}^{1}\right)+\left(\mathrm{C}_{m-6}^{3}+\mathrm{C}_{m-6}^{2}\right)+\cdots= \\
\left(\mathrm{C}_{m-3}^{0}+\mathrm{C}_{m-4}^{1}+\mathrm{C}_{m-5}^{2}+\mathrm{C}_{m-6}^{3}+\cdots\right)+\left(\mathrm{C}_{m-4}^{0}+\mathrm{C}_{m-5}^{1}+\right. \\
\left.\mathrm{C}_{m-6}^{2}+\cdots\right)|f(m-1)|+|f(m-2)|,
\end{array}
$$

Since $|f(2)|=F_{1}=1,|f(3)|=F_{2}=1$, we have $|f(m)|=F_{m-1}$, where the sequence $\left\{F_{n}\right\}$ is the Fibonacci sequence:
$$
\begin{array}{l}
\quad F_{1}=F_{2}=1, F_{n+2}=F_{n+1}+F_{n}, \text { solving } \lambda^{2}=\lambda+1 \text { gives } \lambda \\
=\frac{1 \pm \sqrt{5}}{2}, \alpha_{n}=\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}, \beta_{n}=\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}, \text { solving } \\
\left\{\begin{array}{l}
A+B=1 \\
\frac{1+\sqrt{5}}{2} A+\frac{1-\sqrt{5}}{2} B=1
\end{array} \text { gives } A=\frac{1+\sqrt{5}}{2 \sqrt{5}}, B=\frac{1-\sqrt{5}}{2 \sqrt{5}}, F_{n}\right. \\
=A \alpha_{n}+B \beta_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right], \\
|f(m)|=F_{m-1}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{m-1}-\left(\frac{1-\sqrt{5}}{2}\right)^{m-1}\right] .
\end{array}
$$