Fibonacci: Malsamoj inter versioj

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==Fibonacci-sekvenco==
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''Liber Abaci'' posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as [[Fibonacci number]]s. Although Fibonacci's ''Liber Abaci'' contains the earliest known description of the sequence outside of India, the sequence had been noted by Indian mathematicians as early as the sixth century.<ref>{{cite journal|first=Pamanand|last=Singh|title=The so-called fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|year=1985|pages=229–244|doi=10.1016/0315-0860(85)90021-7}}</ref><ref>{{cite book |title = Toward a Global Science | first = Susantha | last = Goonatilake |publisher = Indiana University Press |year = 1998 |page = 126 |isbn = 978-0-253-33388-9 |url = https://books.google.com/?id=SI5ip95BbgEC&pg=PA126&dq=Virahanka+Fibonacci }}</ref><ref>{{cite book |title = The Art of Computer Programming: Generating All Trees – History of Combinatorial Generation; Volume 4 |first=Donald |last = Knuth |publisher=Addison-Wesley |year=2006 |isbn=978-0-321-33570-8 | page=50 |url=https://books.google.com/?id=56LNfE2QGtYC&pg=PA50&dq=rhythms}}</ref><ref>Hall, Rachel W. [http://www.sju.edu/~rhall/mathforpoets.pdf Math for poets and drummers]. ''Math Horizons'' '''15''' (2008) 10–11.</ref>
 
In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0, 1, 1, 2, as modern mathematicians do but with 1,1, 2, etc. He carried the calculation up to the thirteenth place (fourteenth in modern counting), that is 233, though another manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.<ref>[http://oeis.org/A000045 Fibonacci Numbers] from [http://oeis.org/ The On-Line Encyclopedia of Integer Sequences].</ref><ref>[https://books.google.com/books?id=gvRFAAAAcAAJ&printsec=frontcover&dq=liber+abbaci&hl=en&sa=X&ei=MrGzUfC8LYS-0AG2pYCwDg&ved=0CDAQ6AEwAA#v=snippet&q=3%205%208%2013%2021%2034&f=false ''Il Liber Abbaci'', 1857 edition, p. 231. Online at]</ref> Fibonacci did not speak about the [[golden ratio]] as the limit of the ratio of consecutive numbers in this sequence.
 
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