Fibonacci Calculator
The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… where each number is the sum of the two before it. F(n) = F(n-1) + F(n-2). The ratio of consecutive terms converges to the Golden Ratio φ ≈ 1.618. Appears throughout nature in spirals, petals, and seed arrangements.
Calculate Fibonacci numbers up to F(1000) using BigInt. Generate sequences, check if a number is Fibonacci, find nearest Fibonacci, explore the Golden Ratio, and visualize the Fibonacci spiral.
Enter index n (0–1000)
Golden Ratio (φ) & Fibonacciφ = 1.6180339887498948…
The ratio of consecutive Fibonacci numbers F(n)/F(n−1) converges to the Golden Ratio φ ≈ 1.618…
| n | F(n) | F(n)/F(n−1) | |Diff from φ| |
|---|---|---|---|
| 2 | 1 | 1.000000000000000 | 6.180340e-1 |
| 3 | 2 | 2.000000000000000 | 3.819660e-1 |
| 4 | 3 | 1.500000000000000 | 1.180340e-1 |
| 5 | 5 | 1.666666666666667 | 4.863268e-2 |
| 6 | 8 | 1.600000000000000 | 1.803399e-2 |
| 7 | 13 | 1.625000000000000 | 6.966011e-3 |
| 8 | 21 | 1.615384615384615 | 2.649373e-3 |
| 9 | 34 | 1.619047619047619 | 1.013630e-3 |
| 10 | 55 | 1.617647058823529 | 3.869299e-4 |
| 11 | 89 | 1.618181818181818 | 1.478294e-4 |
| 12 | 144 | 1.617977528089888 | 5.646066e-5 |
| 13 | 233 | 1.618055555555556 | 2.156681e-5 |
| 14 | 377 | 1.618025751072961 | 8.237677e-6 |
| 15 | 610 | 1.618037135278515 | 3.146529e-6 |
Fibonacci in Nature 🌻
Sunflowers
Seed spirals follow Fibonacci numbers (34 and 55 spirals)
Nautilus Shell
Shell chambers grow in a Fibonacci spiral pattern
Leaf Arrangement
Phyllotaxis: leaves spiral at Fibonacci angles (137.5°)
Pineapple Scales
Spirals of 8, 13, and 21 — all Fibonacci numbers
Flower Petals
Lilies (3), buttercups (5), daisies (34 or 55)
Galaxies
Spiral galaxies approximate the golden spiral shape
Quick Reference
| F(0)–F(10) | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 |
| F(20) | 6,765 |
| F(50) | 12,586,269,025 |
| F(100) | 354,224,848,179,261,915,075 (21 digits) |
| F(1000) | 209 digits |
Key Formulas
- F(n) = F(n−1) + F(n−2), F(0)=0, F(1)=1
- Binet: F(n) = (φⁿ − ψⁿ) / √5
- φ = (1 + √5) / 2 ≈ 1.618034…
- ψ = (1 − √5) / 2 ≈ −0.618034…
- lim F(n)/F(n−1) = φ (Golden Ratio)
How to Use
- Enter your value in the input field
- Click the Calculate/Convert button
- Copy the result to your clipboard
Frequently Asked Questions
- What is the Fibonacci sequence?
- The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… It was introduced to Western mathematics by Leonardo of Pisa (Fibonacci) in 1202 in his book Liber Abaci.
- How is the Golden Ratio related to Fibonacci numbers?
- The ratio of consecutive Fibonacci numbers F(n)/F(n-1) converges to the Golden Ratio φ ≈ 1.6180339887… as n increases. By F(10)/F(9) = 89/55 ≈ 1.6182, the ratio is already very close. The exact value is φ = (1 + √5) / 2. This connection is why Fibonacci spirals approximate golden spirals.
- How can I check if a number is a Fibonacci number?
- A positive integer n is a Fibonacci number if and only if 5n² + 4 or 5n² − 4 is a perfect square. For example, 13: 5(169)+4 = 849 (not square), 5(169)−4 = 841 = 29² ✓, so 13 is Fibonacci. This test works for any number without generating the full sequence.
- Where does the Fibonacci sequence appear in nature?
- Fibonacci numbers appear throughout nature: sunflower seed spirals (34 and 55), pinecone spirals (8 and 13), pineapple scales, nautilus shell chambers, leaf arrangements (phyllotaxis at 137.5° angles), and flower petals (lilies have 3, buttercups 5, daisies 34 or 55). This occurs because Fibonacci-based growth optimizes packing and light exposure.
- What is the largest Fibonacci number this calculator supports?
- This calculator computes Fibonacci numbers up to F(1000) using BigInt arbitrary-precision arithmetic. F(1000) has 209 digits. The computation is instant because we use an iterative method with O(n) time complexity rather than the exponential-time recursive approach.
- What is Binet's formula for Fibonacci numbers?
- Binet's formula gives F(n) = (φⁿ − ψⁿ) / √5, where φ = (1+√5)/2 ≈ 1.618 (golden ratio) and ψ = (1−√5)/2 ≈ −0.618. It calculates any Fibonacci number directly without iteration. However, for large n, floating-point precision limits make the iterative method more reliable for exact values.