Fibonacci Sequence Calculator

The Fibonacci sequence is the list of integers defined by F(0) = 0, F(1) = 1, and F(n) = F(n - 1) + F(n - 2) for n at least 2. Every term after the first two is the sum of the two terms before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. The sequence shows up across mathematics, biology, and finance because the recurrence is the simplest non-trivial linear pattern that grows in both directions.

As n grows, the ratio of consecutive Fibonacci numbers F(n) / F(n - 1) approaches the golden ratio phi = (1 + sqrt(5)) / 2, about 1.6180339887. By F(20) the ratio matches phi to seven decimal places, and by F(50) it matches to fifteen. This calculator returns that running approximation so you can watch convergence in real time. The link is no coincidence: phi is the dominant root of the characteristic equation x^2 = x + 1 that the Fibonacci recurrence satisfies.

Binet's closed-form expression is F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1 + sqrt(5)) / 2 and psi = (1 - sqrt(5)) / 2. It returns the exact integer F(n) in theory, but on a computer phi^n is stored as a 64-bit float, and rounding error accumulates as n grows. Around F(71) the rounded result starts to drift by one from the true value, so this calculator only uses Binet up to n = 70 and switches to iterative BigInt arithmetic above that.

Counts of spiral arms on pinecones (often 8 and 13), pineapples (8 and 13), and sunflower heads (commonly 34 and 55, sometimes 89 and 144) line up with consecutive Fibonacci numbers. Nautilus shells grow in a logarithmic spiral whose proportions sit close to the golden ratio. Many plants pack new leaves around a stem at an angle of about 137.5 degrees, the so-called golden angle, which packs the most leaves into the available light without overlap. Fibonacci numbers also appear in the branching of trees and the family trees of honeybee drones.

F(79) is the first Fibonacci number larger than Number.MAX_SAFE_INTEGER (2^53 - 1, about 9 x 10^15). F(78) is 8,944,394,323,791,464, which still fits exactly in a 64-bit double; F(79) is 14,472,334,024,676,221, which does not. Above that limit, a standard JavaScript number cannot represent every integer exactly, so additions silently round and the displayed value becomes wrong. BigInt stores integers of arbitrary size and never rounds, so this calculator computes every term with BigInt internally and only converts back to a number for display when the value still fits safely. Beyond that threshold you see the exact digits as a string.