Fibonacci sequence: Difference between revisions

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=== Magnitude ===

Since ''F_n'' is [[Asymptotic analysis|asymptotic]] to <math>\varphi^n/\sqrt5</math>, the number of digits in {{math|''F''_''n''}} is asymptotic to <math>n\log_{10}\varphi\approx 0.2090\, n</math>. As a consequence, for every integer {{math|''d'' > 1}} there are either 4 or 5 Fibonacci numbers with {{mvar|d}} decimal digits.

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

|}

More generally, in the [[radix|base]] {{mvar|b}} representation, the number of digits in {{math|''F''_''n''}} is asymptotic to <math>n\log_b\varphi = \frac{n \log \varphi}{\log b}.</math>

=== 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>

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* [[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 ==