Fibonacci Calculator
The nth Fibonacci number and the sequence up to it — plus how quickly the ratio of neighbours closes on the golden ratio.
Each Fibonacci number is the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8, 13.
How the fibonacci calculator works
Each Fibonacci number is the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8, 13. The sequence is named for Leonardo of Pisa, who used it in 1202 to model rabbit populations, though Indian mathematicians had described it centuries earlier.
The striking property is the ratio: divide any Fibonacci number by the one before it and the answer converges on φ = 1.6180339887…, alternating above and below as it goes. By the twentieth term the ratio is right to seven decimal places.
Beyond n = 78 the values exceed what a double-precision number can hold exactly, so results are flagged as approximate from there on.
Formula: F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2)
Worked examples
| Inputs | F(n) | Note |
|---|---|---|
| The 20th Fibonacci number | 6,765 | 6,765 |
| The 10th | 55 | 55 |
| Where it starts | 1 | 1 |
FAQFrequently asked questions
What is the Fibonacci sequence?
0, 1, 1, 2, 3, 5, 8, 13, 21 … each number the sum of the two before it.
What is the 20th Fibonacci number?
6,765, counting F(0) = 0 as the start.
Why does the ratio approach 1.618?
Because the sequence grows geometrically, and the only growth rate consistent with F(n) = F(n−1) + F(n−2) is the golden ratio.
Does the sequence start at 0 or 1?
Conventions differ. This page uses F(0) = 0, F(1) = 1, which is the modern standard; Fibonacci himself started at 1, 1.
Where does it appear in nature?
In phyllotaxis — the spiral counts of sunflower seeds, pine cones and pineapple scales are almost always Fibonacci numbers, because that packing is the most efficient.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.