Fibonacci sequence

The Fibonacci sequence is:

$$0,1,1,2,3,5,8,13,21,34,55,89,\ldots$$

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.

It is represented mathematically by stating that the $$n$$-th term is given by,

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

Golden ratio
The ratio of sequential terms converges to the golden ratio of roughly 1.6. Or more formally,

$$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\frac{1+\sqrt5}{2}=\varphi$$

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).