# Fibonacci sequence

The *Fibonacci sequence ^{}*, discovered by Leonardo Pisano Fibonacci, begins

$$0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,\mathrm{\dots}$$ |

(Sequence^{} http://www.research.att.com/projects/OEIS?Anum=A000045A000045 in [1]).
The $n$th Fibonacci number is generated by adding the previous two. Thus, the Fibonacci sequence has the recurrence relation

$${f}_{n}={f}_{n-1}+{f}_{n-2}$$ |

with ${f}_{0}=0$ and ${f}_{1}=1$. This recurrence relation can be solved into the closed form

$${f}_{n}=\frac{1}{\sqrt{5}}\left({\varphi}^{n}-{\varphi}^{\prime n}\right)$$ |

called the *Binet formula*, where $\varphi $ denotes the golden ratio^{} (and ${\varphi}^{\prime}$ is defined in the same entry). Note that

$$\underset{n\to \mathrm{\infty}}{lim}\frac{{f}_{n+1}}{{f}_{n}}=\varphi .$$ |

## References

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

Title | Fibonacci sequence |

Canonical name | FibonacciSequence |

Date of creation | 2013-03-22 11:56:07 |

Last modified on | 2013-03-22 11:56:07 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 21 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 11B39 |

Synonym | Fibonacci number |

Related topic | HogattTheorem |

Related topic | LucasNumbers |

Related topic | ZeckendorfsTheorem |

Related topic | ApplicationsOfSecondOrderRecurrenceRelationFormula |

Defines | Binet formula |