Fibonacci numbers

The sequence is named for Leonardo of Pisa, known as Fibonacci, who presented it to European readers in Liber Abaci in 1202. He framed it as a puzzle about breeding rabbits: starting with one pair that becomes productive after a month, how many pairs exist after a year? Counting the pairs month by month produces 1, 2, 3, 5, 8, 13 and onwards. The book's real argument was not about rabbits at all — Fibonacci was making the case for Hindu–Arabic numerals over Roman ones, and the puzzle was a demonstration of how much easier calculation becomes in a positional system.

The sequence was already old by then. Indian scholars studying Sanskrit prosody had derived it while counting the ways to arrange short and long syllables in a line of fixed duration. Pingala's work on metre, dating to roughly the third or second century BCE, contains the germ of the idea; Virahanka stated the recurrence explicitly around 700 CE, and Gopala and Hemachandra discussed it in the twelfth century, shortly before Liber Abaci appeared.

The modern name is more recent still. The nineteenth-century French mathematician Édouard Lucas attached Fibonacci's name to the sequence while studying its divisibility properties, and also gave his own name to the closely related Lucas numbers, which follow the same rule from a different pair of starting values.

Two results give the sequence its reach. The ratio of consecutive terms converges on the golden ratio φ ≈ 1.6180339887, which is why the numbers keep surfacing in discussions of proportion. And Binet's formula expresses the nth term in closed form using powers of φ, meaning any term can be computed directly without stepping through all its predecessors.

Key properties

• F(0) = 0, F(1) = 1, and F(n) = F(n−1) + F(n−2) for every n greater than 1.
• The ratio F(n+1)/F(n) converges to the golden ratio φ = (1+√5)/2 ≈ 1.6180339887.
• Every third Fibonacci number is even; every fourth is divisible by 3; every fifth by 5.
• gcd(F(m), F(n)) = F(gcd(m, n)) — the sequence preserves greatest common divisors.
• Zeckendorf’s theorem: every positive integer is a unique sum of non-consecutive Fibonacci numbers.
• The only perfect squares in the sequence are 0, 1 and 144.
• F(79) = 14,472,334,024,676,221 exceeds the exact-integer range of a JavaScript number, so this page computes with arbitrary-precision arithmetic.

Where they turn up

• The number of spirals in sunflower heads, pinecones and pineapples is almost always a Fibonacci number, a consequence of the packing angle that gives seeds the most even spacing.
• Fibonacci search and Fibonacci heaps use the sequence to structure comparisons and merges efficiently.
• Traders draw "Fibonacci retracement" lines on price charts at ratios derived from the sequence — a convention of technical analysis rather than a result with predictive support.
• Poetic metre: the original Indian derivation counted syllable arrangements, making this a result from linguistics before it was one from biology.
• The Fibonacci word and related substitution sequences appear in the study of quasicrystals and aperiodic tilings.