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In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,[1]

${\displaystyle F_{0}=0,\quad F_{1}=1,}$

and

${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$

for n > 1.

The beginning of the sequence is thus:

${\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }$[2]

In some older books, the value ${\displaystyle F_{0}=0}$ is omitted, so that the sequence starts with ${\displaystyle F_{1}=F_{2}=1,}$ and the recurrence ${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$n > 1.

The beginning of the sequence is thus:

#### Decomposition of powers

Since the golden ratio satisfies the equation Since the golden ratio satisfies the equation

${\displaystyle \varphi ^{2}=\varphi +1,}$

this expression can be used to decompose higher powers ${\displaystyle \varphi$

this expression can be used to decompose higher powers ${\displaystyle \varphi ^{n}}$ as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of ${\displaystyle \varphi }$ and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:

${\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}This equation can be proved by induction on$n.

This expression is also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule ${\displaystyle F_{n}=F_{n-1}+F_{n-2}.}$extended to negative integers using the Fibonacci rule ${\displaystyle F_{n}=F_{n-1}+F_{n-2}.}$

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is