Editing Fibonacci sequence (section)

===Europe===
[[File:Liber abbaci magliab f124r.jpg|thumb|upright=1.25|A page of [[Fibonacci]]'s {{lang|la|[[Liber Abaci]]}} from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) 13 entries of the Fibonacci sequence:<br /> the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.]]

The Fibonacci sequence first appears in the book {{lang|la|[[Liber Abaci]]}} (''The Book of Calculation'', 1202) by [[Fibonacci]],{{Sfn|Sigler|2002|pp=404–405}}<ref>{{citation|url=https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html|title=Fibonacci's Liber Abaci (Book of Calculation)|date=13 December 2009|website=[[The University of Utah]]|access-date=28 November 2018}}</ref> where it is used to calculate the growth of rabbit populations.<ref>{{citation
 | last = Tassone | first = Ann Dominic
 | date = April 1967
 | doi = 10.5951/at.14.4.0285
 | issue = 4
 | journal = The Arithmetic Teacher
 | jstor = 41187298
 | pages = 285–288
 | title = A pair of rabbits and a mathematician
 | volume = 14}}</ref> Fibonacci considers the growth of an idealized ([[biology|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 rabbit [[Mathematical problem|math problem]]: 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 they produce a new pair, so there are 2 pairs in the field.
* At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.
* At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.

At the end of the {{mvar|n}}-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month {{math|''n'' – 2}}) plus the number of pairs alive last month (month {{math|''n'' – 1}}).  The number in the {{mvar|n}}-th month is the {{mvar|n}}-th Fibonacci number.<ref>{{citation | last = Knott | first = Ron
 | title = Fibonacci's Rabbits | url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[[University of Surrey]] Faculty of Engineering and Physical Sciences}}</ref>

The name "Fibonacci sequence" was first used by the 19th-century number theorist [[Édouard Lucas]].<ref>{{Citation | first = Martin | last = Gardner | author-link = Martin Gardner |title=Mathematical Circus |publisher = The Mathematical Association of America |year=1996 |isbn= 978-0-88385-506-5 | quote = It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas...  attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci | page = 153}}</ref>

[[File:Fibonacci Rabbits.svg|left|thumb|upright=1.5|Solution to Fibonacci rabbit [[Mathematical problem|problem]]: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At ''the end of the n''th month, the number of pairs is equal to ''F<sub>n.</sub>'']]
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