Fibonacci Calculator
Calculate Fibonacci numbers and explore the golden ratio.
Understanding the Fibonacci Sequence
The Fibonacci sequence is one of the most famous sequences in mathematics. Named after Italian mathematician Leonardo of Pisa (known as Fibonacci), it appears throughout nature, art, and architecture. Each number in the sequence is the sum of the two preceding numbers.
The Sequence
Definition: F(n) = F(n-1) + F(n-2)
Starting values: F(0) = 0, F(1) = 1
The sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610...
The Golden Ratio (φ)
As the Fibonacci sequence progresses, the ratio of consecutive numbers approaches the golden ratio (phi, φ):
φ = (1 + √5) / 2 ≈ 1.618033988749895
| n | F(n) | F(n)/F(n-1) |
|---|---|---|
| 5 | 5 | 5/3 = 1.667 |
| 10 | 55 | 55/34 = 1.618 |
| 15 | 610 | 610/377 = 1.6180... |
| 20 | 6765 | 6765/4181 = 1.618033... |
Fibonacci in Nature
The Fibonacci sequence appears remarkably often in nature:
- Flower Petals: Lilies (3), buttercups (5), delphiniums (8), marigolds (13), daisies (21, 34, 55, 89)
- Seed Heads: Sunflower seeds spiral in Fibonacci numbers
- Pinecones: Spirals follow Fibonacci patterns
- Tree Branches: Growth patterns often follow the sequence
- Shells: Nautilus shells approximate the golden spiral
- Hurricanes & Galaxies: Spiral structures related to golden ratio
Mathematical Properties
| Property | Formula/Description |
|---|---|
| Sum of first n | F(1) + F(2) + ... + F(n) = F(n+2) - 1 |
| Every 3rd is even | F(3), F(6), F(9)... are divisible by 2 |
| Every 4th is div by 3 | F(4), F(8), F(12)... are divisible by 3 |
| GCD property | GCD(F(m), F(n)) = F(GCD(m, n)) |
| Binet's Formula | F(n) = (φⁿ - ψⁿ) / √5, where ψ = (1-√5)/2 |
Applications
- Computer Science: Algorithm analysis, data structures
- Financial Markets: Fibonacci retracement levels in trading
- Art & Design: Golden ratio in composition
- Architecture: Proportions in famous buildings
- Music: Timing and structure in compositions
First 20 Fibonacci Numbers
| F(0) = 0 | F(10) = 55 |
| F(1) = 1 | F(11) = 89 |
| F(2) = 1 | F(12) = 144 |
| F(3) = 2 | F(13) = 233 |
| F(4) = 3 | F(14) = 377 |
| F(5) = 5 | F(15) = 610 |
| F(6) = 8 | F(16) = 987 |
| F(7) = 13 | F(17) = 1597 |
| F(8) = 21 | F(18) = 2584 |
| F(9) = 34 | F(19) = 4181 |
Golden Ratio Facts
- φ ≈ 1.618033988749895
- φ² = φ + 1
- 1/φ = φ - 1
- Also called "divine proportion"