TAOCP 1.2.8 Exercise 1
Fibonacci's original problem assumes that a pair of rabbits produces a new pair every month, starting from one newly born pair, and that rabbits become productive after one month.
Section 1.2.8: Fibonacci Numbers
Exercise 1. [10] What is the answer to Leonardo Fibonacci's original problem: How many pairs of rabbits are present after a year?
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Fibonacci's original problem assumes that a pair of rabbits produces a new pair every month, starting from one newly born pair, and that rabbits become productive after one month. Let $F_n$ denote the number of pairs at the end of month $n$. By definition, $F_1 = 1$ and $F_2 = 1$, and for $n \ge 3$ we have $F_n = F_{n-1} + F_{n-2}$. After one year, that is, after 12 months, the number of pairs is
$$ F_{12} = F_{11} + F_{10}. $$
Using the Fibonacci sequence $1,1,2,3,5,8,13,21,34,55,89,144$, we find $F_{12} = 144$.
Therefore, the number of rabbit pairs present after one year is
$$ \boxed{144}. $$