Fibonacci sequence

One common indexing begins with F₀ = 0 and F₁ = 1, then defines each later term by Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2. This gives 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Some accounts omit the initial zero and list the sequence as 1, 1, 2, 3, 5, 8, …. The recurrence is the essential rule: each new value depends on the two immediately before it.

The sequence has many connections to number theory, algebra and combinatorics. A particularly well-known relationship concerns the ratio of adjacent terms: as the terms grow, Fₙ₊₁/Fₙ approaches the golden ratio, (1 + √5)/2, approximately 1.618. The terms themselves are integers and do not become the golden ratio; it is their successive ratios that tend toward this limit. The recurrence also gives rise to identities relating terms at different positions, making the sequence a useful example of how a compact rule can produce rich mathematical structure.

The sequence is associated with Leonardo of Pisa, commonly called Fibonacci. In Liber abaci, completed in 1202, he posed a hypothetical problem asking how rabbit pairs might multiply month by month under specified assumptions. The resulting counts follow the same recurrence. The puzzle is an idealized mathematical illustration, not a general model of animal populations. Liber abaci also helped spread Hindu-Arabic numerals and decimal place-value arithmetic in Europe. The modern name of the sequence reflects Fibonacci's influential presentation, rather than being a description of how the sequence is constructed.

The sequence is a standard example of a recurrence relation: a short rule determines each new term from earlier ones. Similar recurrences arise in counting problems when a set of arrangements can be divided into cases whose counts match the previous two totals. Mathematicians also study Fibonacci numbers through identities, divisibility, and their relationships to other sequences. In any real-world application, the recurrence is a model whose assumptions need to fit the problem being considered.