Fibonacci Sequence

In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

0,;1,;1,;2,;3,;5,;8,;13,;21,;34,;55,;89,;144,; ldots; (sequence A000045 in OEIS)

or, alternatively,[1]

1,;1,;2,;3,;5,;8,;13,;21,;34,;55,;89,;144,; ldots.

By definition, the first two numbers in the Fibonacci sequence are 0 and 1 (alternatively, 1 and 1), and each subsequent number is the sum of the previous two.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

F_n = F_{n-1} + F_{n-2},!,

with seed values[2]

F_0 = 0,; F_1     = 1

in the first form, or

F_1 = 1,; F_2 = 1

in the second form.

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics,[3]although the sequence had been described earlier in Indian mathematics.[4][5][6] (By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber Abaci began the sequence with F1 = 1, without an initial 0.)