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Fibonacci Numbers

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

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Fibonacci Numbers -

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# Fibonacci Numbers

## Fibonacci Numbers Calculator

### Display of the sequence

 Display of the sequence Around the last values Only the last number All the sequence (100 terms max)

### Initial values (seeds)

Default seeds for Fibonacci sequence are 0 and 1.

## Answers to Questions (FAQ)

### What is the Fibonacci sequence? (Definition)

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.

### How to calculate the Fibonacci sequence?

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.

### How to find the nth term of the Fibonacci sequence?

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.

### What are the first terms of the Fibonacci sequence?

Fibonacci sequence first numbers are:

 F(0)= 0 1 1 2 3 5 8 13 21 34 55

For the next Fibonacci terms, use the calculator above.

### How to compute the previous Fibonacci term?

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 $F(9) = 34$

### What is the Fibonacci Rabbits' problem?

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.

### What is the difference between the Fibonacci sequence and the Lucas 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.

### What is the Fibonacci sequence algorithm?

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

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Fibonacci Numbers on dCode.fr [online website], retrieved on 2024-06-14, https://www.dcode.fr/fibonacci-numbers

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