# What is a Fibonacci sequence

## Fibonacci sequence

- Fibonacci sequence (3:27 minutes)

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### introduction

The **Fibonacci sequence** is an infinite sequence of numbers, where the following number is obtained by adding the two previous numbers:

Leonardo Fibonacci used this sequence in 1202 to describe the growth of a rabbit population.

### Recursive formula

The Fibonacci sequence can be calculated using the following recursive formation law and the initial values \ (f_0 \) and \ (f_1 \).

$$ f_0 = 0 \ qquad \ text {and} \ qquad f_1 = 1 $$Each additional number is the sum of its two predecessors:

$$ f_n = f_ {n-1} + f_ {n-2} \ qquad \ text {for} \ qquad n \ geq 2 $$### table

The following table contains the Fibonacci numbers for \ (n \ leq 50 \):

\ (n \) | \ (f_n \) |
---|---|

0 | 0 |

1 | 1 |

2 | 1 |

3 | 2 |

4 | 3 |

5 | 5 |

6 | 8 |

7 | 13 |

8 | 21 |

9 | 34 |

10 | 55 |

11 | 89 |

12 | 144 |

13 | 233 |

14 | 377 |

15 | 610 |

16 | 987 |

17 | 1.597 |

18 | 2.584 |

19 | 4.181 |

20 | 6.765 |

21 | 10.946 |

22 | 17.711 |

23 | 28.657 |

24 | 46.368 |

25 | 75.025 |

26 | 121.393 |

27 | 196.418 |

28 | 317.811 |

29 | 514.229 |

30 | 832.040 |

31 | 1.346.269 |

32 | 2.178.309 |

33 | 3.524.578 |

34 | 5.702.887 |

35 | 9.227.465 |

36 | 14.930.352 |

37 | 24.157.817 |

38 | 39.088.169 |

39 | 63.245.986 |

40 | 102.334.155 |

41 | 165.580.141 |

42 | 267.914.296 |

43 | 433.494.437 |

44 | 701.408.733 |

45 | 1.134.903.170 |

46 | 1.836.311.903 |

47 | 2.971.215.073 |

48 | 4.807.526.976 |

49 | 7.778.742.049 |

50 | 12.586.269.025 |

### swell

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