Fibonacci sequence

The Fibonacci numbers were invented by Leonardo of Pisa (ca. 1200), aka. Fibonacci, to describe the growth of a rabbit population. They are defined by the following equations:

F(1) = 1
F(2) = 1
F(n + 2) = F(n) + F(n + 1).

The first Fibonacci numbers are

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ...

These numbers described the number of pairs in a (somewhat idealized) rabbit population after n months if it was assumed that

the first month there is just one newly born pair,
newly born pairs become productive from their second month on, and
each month every pair begets a new pair.

The term Fibonacci sequence is also applied more generally to any function g where g(n + 2) = g(n) + g(n + 1). These functions are precisely those with the form g(n) = aF(n) + bF(n - 1) for some a, b, so the Fibonacci sequences form a vector space with the functions F(n) and F(n - 1) as a basis.

As was pointed out by Johannes Kepler, the growth rate of the Fibonacci numbers, that is, F(n + 1) / F(n), converges to the golden mean, denoted φ. This is the positive root of the quadratic x² - x - 1, so φ² = φ + 1. If we multiply both sides by φⁿ, we get φⁿ⁺² = φⁿ⁺¹ + φⁿ, so the function φⁿ is a Fibonacci sequence. The negative root of the quadratic, 1 - φ, can be shown to have the same properties, so the two functions φⁿ and (1-φ)ⁿ form another basis for the space.

This gives us a formula for the normal Fibonacci sequence:

    F(n) = φⁿ / √5 - (1-φ)ⁿ / √5

As n goes to infinity, the second term converges to zero, so the Fibonacci numbers approach the exponential φⁿ / √5, hence their convergent ratios. In fact the second term starts out small enough that the Fibonacci numbers can be obtained from the first term alone, by rounding to the nearest integer. Matiyasevich was able to show that they can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.

A generalization of the Fibonacci sequence are the Lucas sequences. One kind can be defined thus:

   L(0) = 0
   L(1) = 1
   L(n+2) = PL(n+1) + QL(n)

where the normal Fibbonaci sequence is the special case where P = Q = 1. Another kind of Lucas Sequence begins with L(0) = 2, L(1) = P. Such sequences have applications in number theory and primality proving.

An interesting use of the Fibbonaci sequence is for converting miles to kilometers. For instance, if you want to know about how many kilometers 5 miles is, take the Fibonaci number (5) and look at the next one (8). 5 miles is about 8 kilometers.