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Fibonacci sequence

The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F (n), where n is the first term in the sequence, the following equation obtains for n = 0, where the first two terms are defined as 0 and 1 by convention:

F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

In some texts, it is customary to use n = 1. In that case, the first two terms are defined as 1 and 1 by default, and therefore:

F (1) = 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

The Fibonacci sequence is named for Leonardo Pisano (also known as Leonardo Pisano or Fibonacci), an Italian mathematician who lived from 1170 - 1250. Fibonacci used the arithmetic series to illustrate a problem based on a pair of breeding rabbits:

"How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?" The result can be expressed numerically as: 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. The branching patterns in trees and leaves, for example, and the distribution of seeds in a raspberry are based on Fibonacci numbers.

A Sanskrit grammarian, Pingala, is credited with the first mention of the sequence of numbers, sometime between the fifth century B.C. and the second or third century A.D. Since Fibonacci introduced the series to Western civilization, it has had a high profile from time to time. In The Da Vinci Code, for example, the Fibonacci sequence is part of an important clue. Another application, the Fibonacci poem, is a verse in which the progression of syllable numbers per line follows Fibonacci's pattern.

The Fibonacci sequence is related to the golden ratio, a proportion (roughly 1:1.6) that occurs frequently throughout the natural world and is applied across many areas of human endeavor. Both the Fibonacci sequence and the golden ratio are used to guide design for architecture, websites and user interfaces, among other things.

This was last updated in October 2016

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