# Fibonacci sequence

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The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers, with the first two numbers being 0 and 1. The numbers are known as Fibonacci numbers.

Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...

Written mathematically, the Fibonacci sequence Fn satisfies:

F0 = 0,F1 = 1 and $F_{n} = F_{n-1} + F_{n-2}, \forall n \geq 2$.

The Fibonacci numbers were first described by the Sanskrit linguist Pingala around the year 450 BC. They arose from the study of meters with long and short syllables.

The first Western mathematician to study this sequence was Leonardo of Pisa, commonly known as Fibonacci, in the consideration of the growth of an idealized rabbit population.

Johannes Kepler discovered that the limit of the ratios between succeeding Fibonacci numbers converges to the golden ratio. Fibonacci numbers also occur in Pascal's triangle and the run-time analysis of Euclid's Algorithm.

## Example of Fibonacci Sequences

• The family trees of honey bees can be depicted using Fibonacci numbers because unmated females will always hatch male young, while mated females will always hatch female young and the Fibonacci sequence has many practical applications in biology, e.g. in the study of reproductive patterns of animals, in the study of spiraling of flowers, etc.