Tool to compute numbers of Fibonacci. Fibonacci sequence is a sequence of integers, each term is the sum of the two previous ones.

Fibonacci Numbers - dCode

Tag(s) : Series

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

The Fibonacci sequence is an infinite mathematical sequence in which each term is the sum of the two previous terms, usually starting with 0 and 1.

The sequence begins like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Numbers from the Fibonacci sequence, noted $ F_n $ ou $ F(n) $, are equal to the addition of the 2 previous terms, they follow the recurrence formula: $$ F(n) = F(n-1) + F(n-2) $$ that can be also written $$ F(n+2) = F(n) + F(n+1) $$

To initiate the sequence, by default, the two first terms are $ F(0) = 0 $ and $ F(1) = 1 $

__Example:__ $ F_2 = F_0+F_1 = 0+1 = 1 $

$ F_3 = F_1+F_2 = 1+1 = 2 $

$ F_{10} = F_8+F_9 $, etc.

Any other values of $ F_0 $ and $ F_1 $ will produce different Fibonacci sequences.

Binet's formula allows you to calculate the nth term without having to use recursion or having to calculate the previous terms. The formula is $$ F(n) = \frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n - \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n $$ and involves the golden ratio.

Fibonacci sequence first numbers are:

F(0)= | 0 |
---|---|

F(1)= | 1 |

F(2)= | 1 |

F(3)= | 2 |

F(4)= | 3 |

F(5)= | 5 |

F(6)= | 8 |

F(7)= | 13 |

F(8)= | 21 |

F(9)= | 34 |

F(10)= | 55 |

For the next Fibonacci terms, use the calculator above.

Each term in the sequence is equal to the previous multiplied by approximately $ \varphi = 1.618 $ (golden number).

__Example:__ $ F(10) = 55 $, $ 55/\varphi \approx 33.99 $ and in fact $ F(9) = 34 $

The rabbits' growth problem is a problem proposed by Leonardo Fibonacci in 1200.

There is a rabbit couple (male + female) and every month a couple breeds and gives birth to a new pair of rabbits which in turn can reproduce itself after 2 months. How many rabbits will be born after X months?

In the beginning there is 1 couple then

1 month | 1 couple |

two months | 2 couples |

three months | 3 couples |

4 months | 5 couples |

5 months | 8 couples |

6 months | 13 couples |

7 months | 21 couples |

8 months | 34 couples |

Each month, the total number of rabbits is equal to the sum of the numbers of the previous two months because it is the number of existing rabbits (the previous month) plus the number of babies born from rabbits couples who have at least two months (hence the number of rabbits 2 months ago). The numbers found are the numbers of the Fibonacci sequence.

The Lucas sequence is similar to the Fibonacci sequence, but it starts with 2 and 1 (instead of 0 and 1). The recurrence formulas are the same.

A source code to program the calculation of Fibonacci numbers by recurrence:`//Pseudo-code`

function fibonacci(n) {

if (n == 0) return 0

if (n == 1) return 1

return fibonacci(n - 1) + fibonacci(n - 2)

}

And an iterative version of the algorithm `//Pseudo-code`

function fibonacci(n) {

if (n == 0) return 0

if (n == 1) return 1

prev = 0

curr = 1

for i from 2 to n {

next = prev + curr

prev = curr

curr = next

}

return curr

}

dCode retains ownership of the "Fibonacci Numbers" source code. Except explicit open source licence (indicated Creative Commons / free), the "Fibonacci Numbers" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Fibonacci Numbers" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Fibonacci Numbers" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Fibonacci Numbers" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Fibonacci Numbers* on dCode.fr [online website], retrieved on 2024-06-14,

- Fibonacci Numbers Calculator
- What is the Fibonacci sequence? (Definition)
- How to calculate the Fibonacci sequence?
- How to find the nth term of the Fibonacci sequence?
- What are the first terms of the Fibonacci sequence?
- How to compute the previous Fibonacci term?
- What is the Fibonacci Rabbits' problem?
- What is the difference between the Fibonacci sequence and the Lucas sequence?
- What is the Fibonacci sequence algorithm?

fibonacci,fibo,sequence,leonardo,rabbit,number,golden,ratio,addition,growth

https://www.dcode.fr/fibonacci-numbers

© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback