The Fibonacci Sequence

In Fibonacci's Liber Abaci, he discussed the following problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Assuming that no rabbits die, the number at the end of each month follows this sequence. (Fibonacci omitted the first "1", assuming that the first pair breeds immediately):
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
This sequence was christened the Fibonacci sequence by Eduard Lucas in 1877 when he used both it and what we call the Lucas sequence to search for Mersenne primes.

This sequence can be produced by using the rule that each term is the sum of the two previous terms. The next few terms after the ones listed above are 377, 610, 987, 1597, 2584, ... .

One of the properties of the Fibonacci sequence (or any variant of it obtained by starting with two other numbers) is that the ratios of consecutive terms approach 1.61803... ((1 + 5)/2), the Golden Ratio. For example, ²⁵⁸⁴/₁₅₉₇ = 1.618033813..., which is the Golden Ratio to six decimal places.

Lucas discovered a relation which allows us to find Fibonacci numbers using Pascal's Triangle. If you look at the diagonals which go through the triangle at about a 30 degree angle, and sum those terms up, you get a Fibonacci number. See the following diagram:

In 1948, Charles Raine was able to connect Fibonacci numbers to Pythagorean triangles. Take any 4 consecutive Fibonacci numbers; the product of the outer terms and twice the produce of the inner terms are the legs of a Pythagorean triangle, and the hypotenuse will be a Fibonacci number. The area of the triangle will be the product of the original four numbers.

There are many other mathematical properties of this sequence. For example, every prime number divides an infinite number of terms of the sequence. Also, if the n^th Fibonacci number is prime, then n itself is prime (except for F₄ = 3). The converse is not necessarily true though.

The Fibonacci sequence is, rather surprisingly, connected with plant growth. For example, Fibonacci numbers often appear in the scales of a fir cone or in the florets of a sunflower. A sunflower usually has two sets of spiral rows with the number of florets (which become seeds) in both sets being a Fibonacci number, For example, 21 and 34 or 89 and 144.