Fibonacci numbers

Published junio 12, 2026 · By NumberWiki

The Fibonacci sequence — 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … — is the most famous integer sequence in mathematics. Each number is the sum of the two before it, a rule so simple a child can continue it, yet it connects to the golden ratio, the geometry of plants, and a surprising amount of deep number theory.

The rule

Start with 0 and 1. Add them to get 1. Add the last two to get 2. Keep going: each term is the sum of the previous two. In symbols, F(0) = 0, F(1) = 1, and F(n) = F(n−1) + F(n−2) for every n ≥ 2. That recurrence generates the whole sequence forever:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, …

You can read several of these as live pages here: 2, 3, 5, 8, 13, 21, 55, 89, and 144 — which is both a Fibonacci number and a perfect square (12²), the only non-trivial square in the entire sequence.

Leonardo of Pisa and the rabbits

The sequence is named for Leonardo of Pisa, known as Fibonacci ("son of Bonacci"), who introduced it to European mathematics in his 1202 book Liber Abaci. He posed it as a puzzle about idealised rabbit breeding: starting with one pair, if every pair produces a new pair each month and pairs become fertile after one month, how many pairs are there after a year? The monthly totals are exactly the Fibonacci numbers.

The rabbits are a toy model, but Fibonacci's larger achievement was the book itself: Liber Abaci was one of the works that brought the Hindu–Arabic decimal numerals — the digits 0–9 we use today — to a Europe still labouring with Roman numerals. The sequence had in fact been described centuries earlier by Indian scholars (Pingala, Virahanka, Hemachandra) in the context of poetic metre, but it was Fibonacci's name that stuck in the West.

The golden ratio

Here is the sequence's most beautiful property. Divide each Fibonacci number by the one before it — 3/2 = 1.5, 5/3 ≈ 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615 — and the ratios close in on a single number:

φ = (1 + √5) ⁄ 2 ≈ 1.6180339887…

This is the golden ratio, φ ("phi"). The further you go, the closer consecutive Fibonacci ratios get to it. The connection runs deeper than a limit: there is a closed-form expression, Binet's formula, that gives the n-th Fibonacci number directly from φ, with no need to add up all the earlier terms —

F(n) = (φⁿ − ψⁿ) ⁄ √5, where ψ = (1 − √5) ⁄ 2.

It looks like it should produce irrational numbers, yet the two irrational pieces always cancel to leave a whole number. NumberWiki shows the digit-at-a -given-position for φ and √5 on its number pages, a small nod to this connection.

Why it appears in nature

Fibonacci numbers turn up remarkably often in living things: the number of petals on many flowers (3, 5, 8, 13, 21…), the spiral counts of seeds in a sunflower head, the bumps on a pinecone, the branching of some plants. This isn't mysticism — it's efficiency. When a plant grows new elements (leaves, seeds, florets) at a constant angle around a stem, the angle that packs them most evenly without overlap is the "golden angle" of about 137.5°, which comes directly from φ. Growth governed by that angle naturally produces Fibonacci counts in the resulting spirals. The sequence appears because it is the arithmetic of optimal packing, not because nature can count.

It's worth being honest that the Fibonacci sequence is also widely over-claimed — in art, architecture, and the stock market it is frequently asserted where the evidence is thin. The genuine appearances in phyllotaxis (leaf arrangement) are real and well understood; many of the cultural claims are pattern-matching after the fact.

Hidden mathematical structure

Beneath the friendly surface, the Fibonacci numbers are astonishingly rich:

Divisibility mirrors the indices. F(m) divides F(n) exactly when m divides n. And the greatest common divisor of two Fibonacci numbers is itself a Fibonacci number: gcd(F(m), F(n)) = F(gcd(m, n)).
Every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers — Zeckendorf's theorem — giving a kind of "Fibonacci base" for the integers.
Sums telescope neatly. The first n Fibonacci numbers add up to F(n+2) − 1, and the sum of their squares, F(1)² + … + F(n)², equals F(n) · F(n+1) — a fact with a lovely visual proof using nested squares.
The Lucas numbers (2, 1, 3, 4, 7, 11, 18, …) follow the same add-the-last-two rule from a different start and are intimately interwoven with the Fibonacci numbers.

Fibonacci on NumberWiki

Numbers in the sequence are tagged Fibonacci — browse them all there — and the closely related Lucas numbers have their own tag. Because Fibonacci membership is rare (only about 30 Fibonacci numbers fall below ten million), the site treats it as an editorially significant property: a Fibonacci page is never "thin." Each one shows the number's factorization, the surrounding sequence context, and how it renders across historical numeral systems.