What Is the Fibonacci Sequence? (2026)

The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on indefinitely. Named after the 13th-century Italian mathematician Leonardo of Pisa (known as Fibonacci), the sequence appears across mathematics, computer science, nature, and design — making it one of the most studied patterns in all of science.

How the sequence is built

The rule is simple: F(n) = F(n − 1) + F(n − 2), with seed values F(0) = 0 and F(1) = 1. Every subsequent term follows directly from that recurrence. The table below shows the first twelve terms (indices 0–11), their values, and their ratio to the previous term — which converges toward the golden ratio (φ ≈ 1.618).

Where Fibonacci numbers appear

In nature, Fibonacci numbers describe the branching of trees, the arrangement of seeds in a sunflower head, and the spiral of a nautilus shell — all examples of growth that follows the same additive rule. In computer science, the sequence is a canonical example for teaching recursion and dynamic programming; a naïve recursive implementation has exponential time complexity, while a memoised or iterative version runs in linear time. In finance, Fibonacci retracement levels (23.6 %, 38.2 %, 61.8 %) are derived from ratios between terms and are widely used in technical analysis, regardless of whether the underlying theory holds up empirically.

Use the Fibonacci Generator to produce any number of terms instantly, copy them as a list, and explore how the ratio between consecutive terms closes in on the golden ratio.