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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]

and

for n > 1.

The beginning of the sequence is thus:

[2]

In some older books, the value is omitted, so that the sequence starts with and the recurrence 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

this expression can be used to decompose higher powers

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