Fibonacci Numbers

Pairs of Rabbits: We start with one pair of young rabbits at the beginning of the month and keep a monthly record of the numbers of pairs of rabbits we have: $[1]$ pairs to start with. However, our rabbits live forever and are incestuous and give birth to two rabbits, one of each sex. At the beginning of the second month the rabbits become old enough to breed. Our record shows $[1,1]$. At the beginning of the third month our original rabbits have 2 offspring and our record shows $[1,1,2]$ for the number of pairs. At the beginning of the fourth month our original pair give birth to two more rabbits and the first offspring pair born a month earlier become old enough to breed and our record shows $[1,1,2,3]$. The fifth month comes around; The original pair of rabbits give birth to yet another pair of rabbits and their first offspring also give birth to two rabbits. The second offspring of the original pair become old enough to breed. Our record is now $[1,1,2,3,5]$ pairs. The six month starts with the original rabbits giving birth to another pair, as do their first offspring and the second offspring have their offspring. At the same time the first grand offspring of the original pair are able to breed: $[1,1,2,3,5,8]$ pairs. Describing which pairs of rabbits are bred or breeding is cumbersome, but we can deduce from the record that there will be as many rabbits as the previous two months' counts summed: $[1,1,2,3,5,8,13,21,34,55,\ldots]$.

This pattern of numbers occurs in nature. The number of spirals on a sunflower or the number of petals on a flower are Fibonacci numbers, to give a couple of examples.

More formally, let $F_1=1$ and $F_2=1$. Then the $n^{th}$ Fibonacci number is given recursively by $F_n=F_{n-2}+F_{n-1}$.

Fibonacci numbers are tabulated as the entries in the $n^{th}$ power of the matrix $ \left( \begin{array}{cc} 1&1\\ 1&0 \end{array}\right). $