PlanetMath: Fibonacci sequence

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Fibonacci sequence

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The Fibonacci sequence, discovered by Leonardo Pisano Fibonacci, begins

$\displaystyle 0, 1, 1, 2, 3, 5 ,8 , 13, 21, 34, 55, 89, 144, 233, 377, \ldots$

(Sequence A000045 in [1]). The th Fibonacci number is generated by adding the previous two. Thus, the Fibonacci sequence has the recurrence relation

$\displaystyle f_n = f_{n-1} + f_{n-2}$

with and . This recurrence relation can be solved into the closed form

$\displaystyle f_n = \frac{1}{\sqrt{5}} \left( \phi^n - \phi'^{\;n} \right)$

called the Binet formula, where $\phi$ denotes the golden ratio (and $\phi'$ is defined in the same entry). Note that

$\displaystyle \lim_{n\rightarrow \infty} \frac{f_{n+1}}{f_n} = \phi.$

Bibliography

1: N. J. A. Sloane, (2004), The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/˜njas/sequences/.

"Fibonacci sequence" is owned by Koro. [ full author list (2) | owner history (1) ]

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Other names:

Fibonacci number

Also defines:

Binet formula

Attachments:: derivation of Binet formula (Derivation) by drini
list of Fibonacci numbers (Example) by cvalente
Fibonacci fraction (Definition) by PrimeFan

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Cross-references: golden ratio, closed form, recurrence relation, generated by, sequence
There are 34 references to this entry.

This is version 16 of Fibonacci sequence, born on 2001-11-04, modified 2007-04-22.
Object id is 665, canonical name is FibonacciSequence.
Accessed 24070 times total.

Classification:

11B39 (Number theory :: Sequences and sets :: Fibonacci and Lucas numbers and polynomials and generalizations)

Pending Errata and Addenda

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