Fibonacci Numbers

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.

Fibonacci sequence first numbers are:

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' 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

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.