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
Fibonacci jigsaw puzzle (Result) by CompositeFan
random Fibonacci sequence (Definition) by PrimeFan

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Cross-references: golden ratio, closed form, recurrence relation, generated by, sequence
There are 42 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 25029 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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