Example 2-2 Fibonacci Sequence.
The Fibonacci sequence is another typical problem in combinatorial mathematics, proposed as follows: There is a pair of rabbits, one male and one female. Assuming that two months after birth, they can produce another pair of rabbits, one male and one female, every month thereafter, how many pairs of rabbits will there be after $n$ months?
Let the number of pairs of rabbits after $n$ months be $F_{n}$, where the number of pairs of rabbits newly born in the current month is $N_{n}$, and the number of pairs of rabbits left from the previous month is $O_{n}$, then we have:
$$
F_{n}=N_{n}+O_{n},
$$

According to the assumption $N_{n}=F_{n-2}$, that is, the number of pairs of rabbits from the $(n-2)$-th month to the $n$-th month all have the ability to reproduce, $O_{n}=F_{n-1}$ - the number of pairs of rabbits from the previous month, hence
$$
F_{n}=F_{n-1}+F_{n-2}, \quad F_{1}=F_{2}=1
$$
$F_{1}=1, F_{2}=1$, that is, in the 1st month and the 2nd month, the number of pairs of rabbits is 1, these are the initial conditions.
Thus, we have the sequence:
$$
\begin{array}{l}
1,1,1+1=2,2+1=3,2+3=5,3+5=8,5+8=13, \\
8+13=21, \\
\cdots
\end{array}
$$

That is, $1,1,2,3,5,8,13,21,34,55,89,144, \cdots$
Similar to the Tower of Hanoi problem, a sequence is generated. However, while $H(n)$ can be given an explicit formula, $F_{n}$ cannot yet, which is precisely the problem to be addressed in Section 2.2.