Fibonacci sequence

{{mathematics}}{{Math-level|level=1}}

The '''Fibonacci sequence''' is:

<math>0,1,1,2,3,5,8,13,21,34,55,89,\ldots</math><ref>[https://www.mathsisfun.com/numbers/fibonacci-sequence.html Fibonacci Sequence], ''Math Is Fun''</ref>

==Calculating==
Each term is obtained by adding the two previous terms (after the arbitrary start of 0,1). The 0 at the beginning is sometimes omitted. It is named after the 13th century Italian mathematician Leonardo of Pisa, also called Fibonacci, though it is known to have been studied in India much earlier. It is a favorite topic of mathematical hobbyists, as well as professional mathematical study.<ref>[https://www.masterclass.com/articles/fibonacci-sequence-formula Fibonacci Sequence Formula: How to Find Fibonacci Numbers], ''MasterClass'', November 8, 2020</ref>

It is represented mathematically by stating that the <math>n</math>-th term is given by,

<math>\begin{cases}a_0=0,a_1=1\\a_n=a_{n-1}+a_{n-2}&n\ge2\end{cases}</math>

===Golden ratio===
The ratio of sequential terms converges to the [[golden ratio]] of roughly 1.6.<ref>[https://www.nationalgeographic.org/media/golden-ratio/ The Golden Ratio], ''National Geographic''</ref> Or more formally,

<math>\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\frac{1+\sqrt5}{2}=\varphi</math>

The explicit formula (Binet's formula) for the ''n''-th Fibonacci number also includes the golden ratio:

<math>a_n = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)</math>

==Trivia==
In its early days, [[RationalWiki]] theoretically used the Fibonacci sequence to determine the next block interval for ne'er-do-wells, starting at one minute, one hour, or one day, depending on how well they did ne'er. RationalWiki dropped this because some admins are dyscalculic and we felt this discriminated against them.

The sequence is also a fairly good approximation of the conversion between miles and kilometres. Take two consecutive numbers and the lower will be miles and the higher will be kilometres. Although it is less accurate at the start as 1 mile to 1 km is a ''very'' rough conversion. However, by the time it gets to 13 and 21 it is pretty accurate as 13 miles is 20.92 km (correct to one significant figure).<ref group=note>This is a fancy way of saying that the conversion between kilometres and miles is close to the golden ratio, φ, mentioned above. 1 mile=1.60934 km, while φ=1.61803. This is only about 0.5% out, which proves the metric system is an Illuminati conspiracy.</ref>

==Fibonacci gallery==
<gallery>
File:Fibonacci cat.jpg|In a Fibonacci spiral, you start with a rectangle partitioned into 2 squares and then in each step add to the rectangle's longest side a square of the same length.
File:Donald Trump Fibonacci spiral.jpg|The Fibonacci spiral is visible all the way from the YUUUGE, down to the very tiny 
hands
.
</gallery>

==See also==
*[[Fun:Fibonacci to the 1000th iteration]] - warning, large article, but really cool in a geek sort of way.

==External links==
*The Fibonacci Sequence entry at the OEIS - [http://oeis.org/A000045 A000045]

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

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[[Category:Mathematics]]
[[Category:Numbers]]

Categories: Mathematics, Numbers