Fibonacci sequence: Difference between revisions

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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''_''m''+1}}.<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>

Fibonacci sequence of first 0 to 28 whose exponents are Fibonacci sequence:

{| class="wikitable" style="text-align:right"

! n !! F_n !! F_{F_n} {{OEIS|A007570}} !! Digits

|-

| 0 || 0 || 0 || 1

|-

| 1 || 1 || 1 || 1

|-

| 2 || 1 || 1 || 1

|-

| 3 || 2 || 1 || 1

|-

| 4 || 3 || 2 || 1

|-

| 5 || 5 || 5 || 1

|-

| 6 || 8 || 21 || 2

|-

| 7 || 13 || 233 || 3

|-

| 8 || 21 || 10,946 || 5

|-

| 9 || 34 || 5,702,887 || 7

|-

| 10 || 55 || 139,583,862,445 || 12

|-

| 11 || 89 || 1,779,979,416,004,714,189 || 19

|-

| 12 || 144 || 555,565,404,224,292,694,404,015,791,808 || 30

|-

| 13 || 233 || 2,211,236,406,303,914,545,699,412,969,744,873,993,387,956,988,653 || 49

|-

| 14 || 377 || 2,746,979,206,949,941,983,182,30...,998,135,811,615,145,905,740,557 || 79

|-

| 15 || 610 || 13,582,369,791,278,266,616,906,2...,870,247,715,060,398,192,490,855 || 128

|-

| 16 || 987 || 83,428,786,095,010,233,039,452,8...,203,909,274,364,258,193,729,858 || 206

|-

| 17 || 1,597 || 2,533,824,184,645,097,323,735,32...,932,445,129,029,157,243,251,977 || 334

|-

| 18 || 2,584 || 472,691,076,546,530,031,780,685,...,076,177,725,139,767,446,193,963 || 540

|-

| 19 || 4,181 || 2,678,174,576,245,943,735,375,49...,196,409,694,930,324,829,641,381 || 874

|-

| 20 || 6,765 || 2,830,748,520,089,123,910,580,48...,613,343,312,104,170,163,349,890 || 1,414

|-

| 21 || 10,946 || 16,952,165,127,657,070,069,126,3...,862,097,770,881,297,257,812,353 || 2,288

|-

| 22 || 17,711 || 107,302,901,410,621,848,261,304,...,866,972,919,879,335,166,641,889 || 3,702

|-

| 23 || 28,657 || 4,067,444,553,773,051,400,817,55...,514,294,119,121,744,055,455,137 || 5,989

|-

| 24 || 46,368 || 975,928,742,637,524,667,034,181,...,031,729,656,663,323,972,447,616 || 9,690

|-

| 25 || 75,025 || 8,876,152,444,949,440,226,201,15...,931,920,374,703,557,334,525,025 || 15,679

|-

| 26 || 121,393 || 19,369,921,626,318,966,187,122,8...,653,571,176,771,808,239,962,913 || 25,370

|-

| 27 || 196,418 || 384,448,010,820,695,855,556,132,...,820,927,828,000,174,246,945,409 || 41,049

|-

| 28 || 317,811 || 16,651,389,657,924,218,778,521,4...,467,273,617,280,084,005,430,914 || 66,419

|}

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''_''m''+1}}) is obtained by adding one [S] to the {{math|''F''_''m''}} cases and one [L] to the {{math|''F''_''m''−1}} 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=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><ref name="HistoriaMathematica"/>