Fibonacci Calculator
Generate Fibonacci numbers and find the nth term in the sequence.
Enter a whole number from 0 to 5000.
Sequence
The Fibonacci Sequence
The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding ones. It is defined by F(0) = 0, F(1) = 1, and F(n) = F(n−1) + F(n−2) for n ≥ 2. The sequence begins 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 Western mathematics in his 1202 book Liber Abaci, though the sequence had appeared earlier in Indian mathematics.
The Golden Ratio Connection
As the sequence grows, the ratio of consecutive Fibonacci numbers F(n+1) / F(n) approaches the golden ratio φ ≈ 1.6180339887. This deep link means a Fibonacci number can also be approximated with Binet's formula, F(n) ≈ φⁿ / √5, rounded to the nearest integer.
Fibonacci Numbers in Nature
Fibonacci numbers appear throughout nature: in the spiral arrangement of leaves and seeds (phyllotaxis), the branching of trees, the number of petals on many flowers, the spirals of pinecones and sunflowers, and the proportions of nautilus shells. These patterns arise because they allow efficient packing and growth.