# Fibonacci number

A tiling with squares whose side lengths are successive Fibonacci numbers

In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,[1]

${\displaystyle F_{0}=0,\quad F_{1}=1,}$

and

${\displaystyle F_{n}=F_{n-1}+F_{n-2},}$

for n > 1.

One has F2 = 1. In some books, and particularly in old ones, F0, the "0" is omitted, and the Fibonacci sequence starts with F1 = F2 = 1.[2][3] The beginning of the sequence is thus:

${\displaystyle (0,)\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }$[4]
The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling;[5] this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13 and 21.

Fibonacci numbers are 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 named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. They appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics,[6] although the sequence had been described earlier in Indian mathematics.[7][8][9]

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.

Fibonacci numbers are also closely related to Lucas numbers ${\displaystyle L_{n}}$ in that they form a complementary pair of Lucas sequences ${\displaystyle U_{n}(1,-1)=F_{n}}$ and ${\displaystyle V_{n}(1,-1)=L_{n}}$. Lucas numbers are also intimately connected with the golden ratio.

## History

Thirteen ways of arranging long and short syllables in a cadence of length six. Five end with a long syllable and eight end with a short syllable.

The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985.[8][10][11] 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.[9]

Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases. [12] Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD).[13][7] However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):[11]

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].[a]

Hemachandra (c. 1150) is credited with knowledge of the sequence as well,[7] writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."[15][16]

A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.
The number of rabbit pairs form the Fibonacci sequence

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Fibonacci.[6][17] using it to calculate the growth of rabbit populations.[18][19] Fibonacci considers the growth of a hypothetical, idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. Fibonacci posed the puzzle: how many pairs will there be in one year?

• At the end of the first month, they mate, but there is still only 1 pair.
• At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
• At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
• At the end of the fourth month, the original female has produced yet another new pair, and the female born two months ago also produces her first pair, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (that is, the number of pairs in month n − 2) plus the number of pairs alive last month (that is, n − 1). This is the nth Fibonacci number.[20]

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[21]

Other Languages
беларуская: Лікі Фібаначы
Esperanto: Fibonaĉi-nombro
ગુજરાતી: ફિબોનાકિ
한국어: 피보나치 수
Արեւմտահայերէն: Ֆիպոնաչչիի թիւ
Bahasa Indonesia: Bilangan Fibonacci
íslenska: Fibonacci-runa
македонски: Фибоначиева низа
Bahasa Melayu: Nombor Fibonacci
Nederlands: Rij van Fibonacci
norsk nynorsk: Fibonaccifølgja
oʻzbekcha/ўзбекча: Fibonachchi sonlari
Qaraqalpaqsha: Fibonachchi sanları
Simple English: Fibonacci number
slovenščina: Fibonaccijevo število
српски / srpski: Фибоначијев низ
srpskohrvatski / српскохрватски: Fibonačijev niz
svenska: Fibonaccital
українська: Числа Фібоначчі
Tiếng Việt: Dãy Fibonacci
West-Vlams: Reke van Fibonacci