What is the Fibonacci sequence?
The Fibonacci sequence is a list of numbers where each number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on. It is named after Leonardo of Pisa, known as Fibonacci, who introduced it to Europe in 1202 with a puzzle about breeding rabbits. Centuries earlier, Indian mathematicians such as Pingala, Virahanka and Hemachandra had already described the same numbers when counting the rhythms of Sanskrit poetry.
How to use the generator
- Choose how many terms you want, up to 1,000. Every term is calculated exactly, even when it has hundreds of digits.
- Choose whether the sequence starts at 0 or at 1. Both conventions are common.
- Copy the sequence, or type any number in the checker to see whether it is a Fibonacci number and its position.
The formula
F(0) = 0, F(1) = 1
F(n) = F(n − 1) + F(n − 2)
There is also a direct formula, called Binet's formula, using the golden ratio φ: F(n) ≈ φⁿ ÷ √5, rounded to the nearest whole number.
Fibonacci and the golden ratio
Divide any Fibonacci number by the one before it and the answer gets closer and closer to 1.6180339887…, the golden ratio φ. For example, 89 ÷ 55 = 1.61818 and 144 ÷ 89 = 1.61797. The "Ratio of last two terms" box shows this. The golden ratio appears in art, architecture and design, and approximately in nature: the spirals of sunflower seeds and pine cones often come in Fibonacci numbers such as 34 and 55.
How to tell if a number is Fibonacci
A whole number x is a Fibonacci number exactly when 5x² + 4 or 5x² − 4 is a perfect square. This checker simply generates the sequence until it reaches or passes your number, which is fast because the terms grow exponentially.
Frequently asked questions
What is the 100th Fibonacci number?
With F(0) = 0, F(100) = 354,224,848,179,261,915,075. Set the number of terms to 101 to see it.
Where is the Fibonacci sequence used?
In computer algorithms (Fibonacci search and heaps), agile project estimation, trading chart analysis and many maths puzzles.
Are there negative Fibonacci numbers?
The sequence can be extended backwards, giving alternating signs: F(−1) = 1, F(−2) = −1, F(−3) = 2.