In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.^[2][3]

The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself. In mathematical terms, it is defined by the following recurrence relation:

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That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as F_n, for n = 0, 1, 2, … ,20 are:^[4][5]

F0	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12	F13	F14	F15	F16	F17	F18	F19	F20
0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610	987	1597	2584	4181	6765

Every 3rd number of the sequence is even and more generally, every kth number of the sequence is a multiple of F_k.

The sequence extended to negative index n satisfies F_n = F_n−1 + F_n−2 for all integers n, and F_−n = (−1)ⁿ⁺¹F_n:

.., −8, 5, −3, 2, −1, 1, followed by the sequence above.

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