Editing Fibonacci sequence (section)

===India===
{{see also|Golden ratio#History}}
[[File:Fibonacci Sanskrit prosody.svg|thumb|Thirteen ({{math|''F''<sub>7</sub>}}) ways of arranging long and short syllables in a cadence of length six. Eight ({{math|''F''<sub>6</sub>}}) end with a short syllable and five ({{math|''F''<sub>5</sub>}}) end with a long syllable.]]

The Fibonacci sequence appears in [[Indian mathematics]], in connection with [[Sanskrit prosody]].<ref name="HistoriaMathematica">{{Citation|first=Parmanand|last=Singh|title= The So-called Fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|issue=3|pages=229–244|year=1985|doi = 10.1016/0315-0860(85)90021-7|doi-access=free}}</ref><ref name="knuth-v1">{{Citation|title=The Art of Computer Programming|volume=1|first=Donald|last=Knuth| author-link =Donald Knuth |publisher=Addison Wesley|year=1968|isbn=978-81-7758-754-8|url=https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100|page=100|quote=Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns&nbsp;... both Gopala (before 1135&nbsp;AD) and Hemachandra (c.&nbsp;1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)&nbsp;...}}</ref>{{sfn|Livio|2003|p=197}} 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 {{mvar|m}} units is {{math|''F''<sub>''m''+1</sub>}}.<ref name="Donald Knuth 2006 50">{{Citation|title = The Art of Computer Programming | volume = 4. Generating All Trees – History of Combinatorial Generation | first = Donald | last = Knuth | author-link = Donald Knuth |publisher= Addison–Wesley |year= 2006 | isbn= 978-0-321-33570-8 | page = 50 | url= https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 | quote = it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats.  ... there are exactly Fm+1 of them.  For example the 21 sequences when {{math|1=''m'' = 7}} are: [gives list].  In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)}}</ref>

Knowledge of the Fibonacci sequence was expressed as early as [[Pingala]] ({{circa}}&nbsp;450&nbsp;BC–200&nbsp;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 {{mvar|m}} beats ({{math|''F''<sub>''m''+1</sub>}}) is obtained by adding one [S] to the {{math|''F''<sub>''m''</sub>}} cases and one [L] to the {{math|''F''<sub>''m''−1</sub>}} cases.<ref>{{Citation | last = Agrawala | first = VS | year = 1969 | title = ''Pāṇinikālīna Bhāratavarṣa'' (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan | quote = SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala  [1969,  463–76],  after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC}}</ref> [[Bharata Muni]] also expresses knowledge of the sequence in the ''[[Natya Shastra]]'' (c.&nbsp;100&nbsp;BC–c.&nbsp;350&nbsp;AD).<ref name="HistoriaMathematica"/><ref name=GlobalScience>{{Citation|title=Toward a Global Science|first=Susantha|last=Goonatilake|author-link=Susantha Goonatilake|publisher=Indiana University Press|year=1998|page=126|isbn=978-0-253-33388-9|url=https://books.google.com/books?id=SI5ip95BbgEC&pg=PA126}}</ref>
However, the clearest exposition of the sequence arises in the work of [[Virahanka]] (c.&nbsp;700 AD), whose own work is lost, but is available in a quotation by Gopala (c.&nbsp;1135):{{sfn|Livio|2003|p=197}}

<blockquote>Variations of two earlier meters [is the variation]&nbsp;...  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]&nbsp;... In this way, the process should be followed in all ''mātrā-vṛttas'' [prosodic combinations].{{efn|"For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier—three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven [[Mora (linguistics)|morae]] [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <ref>{{Citation|last=Velankar|first=HD|year=1962|title='Vṛttajātisamuccaya' of kavi Virahanka|publisher=Rajasthan Oriental Research Institute|location=Jodhpur|page=101}}</ref>}}</blockquote>

[[Hemachandra]] (c.&nbsp;1150) is credited with knowledge of the sequence as well,<ref name=GlobalScience/> writing that "the sum of the last and the one before the last is the number&nbsp;... of the next mātrā-vṛtta."{{sfn|Livio|2003|p=197–198}}<ref>{{citation|last1=Shah|first1=Jayant|year=1991|title=A History of Piṅgala's Combinatorics|url=https://web.northeastern.edu/shah/papers/Pingala.pdf|publisher=[[Northeastern University]]|page=41|access-date=4 January 2019}}</ref>