The Fibonacci sequence calculator generates the sequence, tracks golden ratio convergence, highlights prime Fibonacci numbers, and exports results. Each term is the sum of the two preceding it, starting from 0 and 1.
Generate Sequence
Sequence
Golden Ratio Convergence
φ = (1 + √5) / 2 ≈ 1.6180339887…
| n | F(n) | F(n)/F(n-1) | Error from φ |
|---|
How to Use the Fibonacci Sequence Calculator
The Fibonacci sequence appears throughout mathematics, nature, art, and finance. This calculator generates any length sequence and reveals the mathematical properties that make Fibonacci numbers fascinating.
Step 1: Set the Number of Terms
Enter how many Fibonacci numbers to generate (1–100). The default is 20, producing: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.
Step 2: Optionally Change Starting Values
The classic sequence starts with 0, 1. You can enter any two starting values for a generalized Fibonacci sequence (also called a Lucas sequence when starting with 2, 1). The calculator maintains the same recursive relationship: each term equals the sum of the previous two.
Step 3: Explore Golden Ratio Convergence
The ratio F(n)/F(n-1) converges to the golden ratio φ ≈ 1.6180339887. Watch this convergence in the table: 5/3 = 1.667, 8/5 = 1.600, 13/8 = 1.625, 21/13 = 1.615, 233/144 = 1.61806, and so on. Each step halves the error from φ approximately.
Prime Fibonacci Numbers
Yellow-highlighted numbers in the grid are prime Fibonacci numbers: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229. Every prime Fibonacci F(p) has a prime index p (with one exception: F(4)=3). Whether infinitely many prime Fibonacci numbers exist remains an open problem in mathematics.
Export Options
Copy the sequence as a comma-separated list for use in spreadsheets or code. Download as CSV to import into Excel, Python, or any data analysis tool. The CSV includes index, value, and ratio columns.
FAQ
What is the Fibonacci sequence?
The Fibonacci sequence is a series where each number is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Named after Leonardo Fibonacci, it appears throughout nature in spirals of shells, flower petals, and tree branching patterns.
What is the golden ratio and how does it relate to Fibonacci?
The golden ratio φ (phi) ≈ 1.6180339887. As the Fibonacci sequence progresses, the ratio of consecutive terms F(n)/F(n-1) converges to φ. For example: 5/3 = 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615, ... approaching 1.618 ever more closely.
What is Binet's formula?
Binet's formula gives the nth Fibonacci number directly: F(n) = (φⁿ - ψⁿ)/√5, where φ = (1+√5)/2 ≈ 1.618 and ψ = (1-√5)/2 ≈ -0.618. This allows computing any Fibonacci number without iterating through the sequence, though it requires arbitrary precision for large n.
Which Fibonacci numbers are prime?
The first few prime Fibonacci numbers are: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229. It's an open problem whether there are infinitely many prime Fibonacci numbers. A Fibonacci number F(n) can only be prime if n itself is prime (with the exception of F(4)=3).
Is this tool free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All computations happen locally in your browser. Nothing is sent to any server.
How many Fibonacci terms can I generate?
This calculator supports up to 100 terms. For large terms (above ~75), BigInt is used internally to ensure exact integer results rather than floating-point approximations.