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The Fibonacci sequence is a series of numbers in which each number is the sum of the two that precede it. Starting at 0 and 1, the first 10 numbers of the sequence look like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on forever. The Fibonacci sequence can be described using a mathematical equation: Xn+2= Xn+1 + Xn

The first thing to know is that the sequence is not originally Fibonacci's, who in fact never went by that name. The Italian mathematician who we call Leonardo Fibonacci was born around 1170, and originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University.

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Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of "Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World," (Princeton University Press, 2017). Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it in 200 B.C. predating Leonardo of Pisa by centuries.

"Liber Abaci" first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century, when mathematicians worked out more about the sequence's mathematical properties. In 1877, French mathematician douard Lucas officially named the rabbit problem "the Fibonacci sequence," Devlin said.

It's true that the Fibonacci sequence is tightly connected to what's now known as the golden ratio, phi, an irrational number that has a great deal of its own dubious lore. The ratio of successive numbers in the Fibonacci sequence gets ever closer to the golden ratio, which is 1.6180339887498948482...

Perhaps the most famous example of all, the seashell known as the nautilus, does not in fact grow new cells according to the Fibonacci sequence, he added. When people start to draw connections to the human body, art and architecture, links to the Fibonacci sequence go from tenuous to downright fictional.

The Fibonacci sequence is a set of integers (the Fibonacci numbers) that starts with a zero, followed by a one, then by another one, and then by a series of steadily increasing numbers. The sequence follows the rule that each number is equal to the sum of the preceding two numbers.

The sequence can theoretically continue to infinity, using the same formula for each new number. Some resources show the Fibonacci sequence starting with a one instead of a zero, but this is fairly uncommon.

The Fibonacci sequence can be calculated mathematically. In this approach, each number in the sequence is considered a term, which is represented by the expression Fn. The n reflects the number's position in the sequence, starting with zero. For example, the sixth term is referred to as F5, and the seventh term is referred to as F6.

The first two equations are essentially stating that the term in the first position equals 0 and the term in the second position equals 1. The third equation is a recursive formula, which means that each number of the sequence is defined by using the preceding numbers. For example, to define the fifth number (F4), the terms F2 and F3 must already be defined. These two numbers, in turn, require that the numbers preceding them are already defined. The numbers continuously build on each other throughout the sequence.

You can try out the formula for yourself, using the table to find the sequence numbers preceding the target term value. For example, the following calculation finds the Fibonacci number for the term in the tenth position (F9):

The challenge with a recursive formula is that it always relies on knowing the previous Fibonacci numbers in order to calculate a specific number in the sequence. For example, you can't calculate the value of the 100th term without knowing the 98th and 99th terms, which requires that you know all the terms before them. There are other equations that can be used, however, such as Binet's formula, a closed-form expression for finding Fibonacci sequence numbers. Another option it to program the logic of the recursive formula into application code such as Java, Python or PHP and then let the processor do the work for you.

The Fibonacci sequence is named for Leonardo Pisano (also known Fibonacci), an Italian mathematician who lived from 1170 to 1250. Fibonacci considered the sequence to be an answer to the following question:

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.

Fibonacci numbers can also be used to define a spiral and 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 reflect the Fibonacci sequence.

The Fibonacci sequence is often associated with 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.

The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.[2][3][4] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.[5]

Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[3][9][10] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm+1.[4]

The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci[16][17] where it is used to calculate the growth of rabbit populations.[18][19] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?

Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli:[22]

If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:

If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.[62] The lengths of the periods for various n form the so-called Pisano periods.[63] Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.

Fibonacci sequences appear in biological settings,[76] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[77] the flowering of artichoke, the arrangement of a pine cone,[78] and the family tree of honeybees.[79][80] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.[81] Field daisies most often have petals in counts of Fibonacci numbers.[82] In 1830, K. F. Schimper and A. Braun discovered that the parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.[83]

It has been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence.[89] A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome ( F 1 = 1 {\displaystyle F_{1}=1} ), and at his parents' generation, his X chromosome came from a single parent ( F 2 = 1 {\displaystyle F_{2}=1} ). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome ( F 3 = 2 {\displaystyle F_{3}=2} ). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome ( F 4 = 3 {\displaystyle F_{4}=3} ). Five great-great-grandparents contributed to the male descendant's X chromosome ( F 5 = 5 {\displaystyle F_{5}=5} ), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.) 17dc91bb1f