In mathematics, the Fibonacci numbers form a sequence defined recursively by:

That is, after two starting values, each number is the sum of the two preceding numbers.

The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.

Using ⁠${\displaystyle F_{n-2}=F_{n}-F_{n-1}}$⁠, one can extend the Fibonacci numbers to negative integers. So we get:

and ⁠${\displaystyle F_{-n}=(-1)^{n+1}F_{n}}$⁠.

See also Negafibonacci coding.

There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio φ, and are based on Binet's formula

The analytic function