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# Fibonacci numbers

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The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of *filius Bonacci*, *son of Bonaccio*). Fibonacci's 1202 book *Liber Abaci* introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.

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## Recurrence equation

The **Fibonacci numbers** are defined by the following homogeneous linear recurrence of order 2 and signature (1, 1) (Cf. Category:Recurrence, linear, order 02, (1,1))

which gives the sequence (Cf. A000045)

- {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, ...}

## Binet's closed-form formula

where

- ,

being the Golden ratio, are the roots of

Since

implies

notice how it compares with the recurrence for the **Fibonacci numbers**

## Limit of consecutive quotients

Johannes Kepler observed that the ratio of consecutive **Fibonacci numbers** 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 almostâ€ť, and concluded that the limit approaches the golden ratio