Fibonacci sequence: Difference between revisions

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Under some older definitions, the value <math>F_0 = 0</math> is omitted, so that the sequence starts with <math>F_1=F_2=1,</math> and the recurrence <math>F_n=F_{n-1} + F_{n-2}</math> is valid for {{math|''n'' > 2}}.{{Sfn | Beck | Geoghegan | 2010}}{{Sfn | Bóna | 2011 | p=180}} <!--Fibonacci started the sequence with index 0: {{math|₀→1, ₁→2, ₂→3, ..., ₁₂→377}}.<ref>{{citation |last1=Leonardo da Pisa |title=File:Liber abbaci magliab f124r.jpg - Wikimedia Commons |date=1202 |url=https://commons.wikimedia.org/wiki/File:Liber_abbaci_magliab_f124r.jpg |language=en}}</ref>-->

The first 10020 Fibonacci numbers {{math|''F_n''}} are:

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

! {{math|''nF''₀}}

! {{math|''F''_n1}}

! {{math|''nF''₂}}

! {{math|''F''_n3}}

! {{math|''nF''₄}}

! {{math|''F''_n5}}

! {{math|''nF''₆}}

! {{math|''F''_n7}}

! {{math|''F''₈}}

! {{math|''F''₉}}

! {{math|''F''₁₀}}

! {{math|''F''₁₁}}

! {{math|''F''₁₂}}

! {{math|''F''₁₃}}

! {{math|''F''₁₄}}

! {{math|''F''₁₅}}

! {{math|''F''₁₆}}

! {{math|''F''₁₇}}

! {{math|''F''₁₈}}

! {{math|''F''₁₉}}

|-

| 0

| 0

| 25

| 75,025

| 50

| 12,586,269,025

| 75

| 2,111,485,077,978,050

|-

| 1

| 1

| 26

| 121,393

| 51

| 20,365,011,074

| 76

| 3,416,454,622,906,707

|-

| 2

| 1

| 27

| 196,418

| 52

| 32,951,280,099

| 77

| 5,527,939,700,884,757

|-

| 3

| 2

| 28

| 317,811

| 53

| 53,316,291,173

| 78

| 8,944,394,323,791,464

|-

| 4

| 3

| 29

| 514,229

| 54

| 86,267,571,272

| 79

| 14,472,334,024,676,221

|-

| 5

| 5

| 30

| 832,040

| 55

| 139,583,862,445

| 80

| 23,416,728,348,467,685

|-

| 6

| 8

| 31

| 1,346,269

| 56

| 225,851,433,717

| 81

| 37,889,062,373,143,906

|-

| 7

| 13

| 32

| 2,178,309

| 57

| 365,435,296,162

| 82

| 61,305,790,721,611,591

|-

| 8

| 21

| 33

| 3,524,578

| 58

| 591,286,729,879

| 83

| 99,194,853,094,755,497

|-

| 9

| 34

| 34

| 5,702,887

| 59

| 956,722,026,041

| 84

| 160,500,643,816,367,088

|-

| 10

| 55

| 35

| 9,227,465

| 60

| 1,548,008,755,920

| 85

| 259,695,496,911,122,585

|-

| 11

| 89

| 36

| 14,930,352

| 61

| 2,504,730,781,961

| 86

| 420,196,140,727,489,673

|-

| 12

| 144

| 37

| 24,157,817

| 62

| 4,052,739,537,881

| 87

| 679,891,637,638,612,256

|-

| 13

| 233

| 38

| 39,088,169

| 63

| 6,557,370,319,842

| 88

| 1,100,087,778,366,101,931

|-

| 14

| 377

| 39

| 63,245,986

| 64

| 10,610,209,857,723

| 89

| 1,779,979,416,004,714,189

|-

| 15

| 610

| 40

| 102,334,155

| 65

| 17,167,680,177,565

| 90

| 2,880,067,194,370,816,120

|-

| 16

| 987

| 411597

| 2584

| 165,580,141

| 664181

| 27,777,890,035,288

| 91

| 4,660,046,610,375,530,309

|-

| 17

| 1,597

| 42

| 267,914,296

| 67

| 44,945,570,212,853

| 92

| 7,540,113,804,746,346,429

|-

| 18

| 2,584

| 43

| 433,494,437

| 68

| 72,723,460,248,141

| 93

| 12,200,160,415,121,876,738

|-

| 19

| 4,181

| 44

| 701,408,733

| 69

| 117,669,030,460,994

| 94

| 19,740,274,219,868,223,167

|-

| 20

| 6,765

| 45

| 1,134,903,170

| 70

| 190,392,490,709,135

| 95

| 31,940,434,634,990,099,905

|-

| 21

| 10,946

| 46

| 1,836,311,903

| 71

| 308,061,521,170,129

| 96

| 51,680,708,854,858,323,072

|-

| 22

| 17,711

| 47

| 2,971,215,073

| 72

| 498,454,011,879,264

| 97

| 83,621,143,489,848,422,977

|-

| 23

| 28,657

| 48

| 4,807,526,976

| 73

| 806,515,533,049,393

| 98

| 135,301,852,344,706,746,049

|-

| 24

| 46,368

| 49

| 7,778,742,049

| 74

| 1,304,969,544,928,657

| 99

| 218,922,995,834,555,169,026

|}

Line 393 ⟶ 201:

=== Limit of consecutive quotients ===

[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers [[convergent sequence|converges]]. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio <math>{{tmath|\varphi\colon </math> }}:<ref>{{Citation|last=Kepler |first=Johannes |title=A New Year Gift: On Hexagonal Snow |year=1966 |isbn=978-0-19-858120-8 |publisher=Oxford University Press |page= 92}}</ref><ref>{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}</ref>

<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>

Line 980 ⟶ 788:

* [[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}

* [[Joseph Schillinger]] (1895–1943) developed [[Schillinger System|a system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.

* In [[software development]], Fibonacci numbers are often used by [[Agile management|agile]] teams operating under the [[Scrum (software development)|Scrum]] framework to size their [[product backlog]] items.<ref>{{cite web |last1=Kathuria |first1=Madhur |title=A Guide to Using the Fibonacci Sequence in Scrum |url=https://resources.scrumalliance.org/Article/guide-using-fibonacci-sequence-scrum |website=Scrum Alliance |access-date=8 August 2025}}</ref>

== See also ==