A sequence born from rabbits

In 1202, Leonardo of Pisa — known as Fibonacci — posed a rabbit problem in his Liber Abaci. A pair of rabbits becomes adult after one month, then produces a new pair every following month. How many pairs after a year? The answer reveals a sequence that now bears his name.

The Fibonacci sequence is defined by recurrence: each term is the sum of the two before it.

   F(0) = 0
   F(1) = 1
   F(n) = F(n-1) + F(n-2)     (for n >= 2)

This unrolls into the first terms:

   n     0  1  2  3  4  5  6   7   8   9   10
   F(n)  0  1  1  2  3  5  8   13  21  34  55

The rabbit count follows exactly this rule. Each month the adults from the previous month are still there, and each of them yields a new pair of babies:

   month     1  2  3  4  5  6
   adults    1  1  2  3  5  8     (= the total of the previous month)
   babies    0  1  1  2  3  5     (= the adults of the previous month)
   total     1  2  3  5  8  13    (adults + babies)

Since a month's total is made of the adults (the previous total) and the babies (the total from two months earlier), we recover "the sum of the two before".

This sequence shows up everywhere in nature: spirals of pine cones, of sunflowers, of shells, the arrangement of leaves around a stem. It is no accident: adding the two previous steps is one of the simplest growth patterns, and living things exploit it. Keep the essential in mind: Fibonacci is "the sum of the two before" — a tiny rule that hides a surprising richness.