Fibonacci Calculator
Calculate Fibonacci numbers, generate sequences, and explore the golden ratio
Calculate Fibonacci Numbers
Position of the Fibonacci number to calculate (supports negative values)
Fibonacci Number Result
(Golden ratio: 1.618034...)
Formula used: Binet's formula: Fₙ = (φⁿ - ψⁿ) / √5
Constants: φ = 1.618034 (Golden ratio), ψ = -0.618034
Fibonacci Sequence
Starting values: F₀ = 0, F₁ = 1
Each term is the sum of the two preceding terms: Fₙ = Fₙ₋₁ + Fₙ₋₂
Golden Ratio (φ)
The ratio of consecutive Fibonacci numbers approaches the golden ratio:
This convergence happens remarkably quickly, reaching high precision by F₁₀/F₉.
Fibonacci Properties
Every 3rd number is even
Every 4th number is divisible by 3
Every 5th number is divisible by 5
Sum of first n terms: Σ(Fᵢ) = Fₙ₊₂ - 1
Gcd(Fₘ, Fₙ) = F(gcd(m,n))
Famous Fibonacci Numbers
Understanding the Fibonacci Sequence
What is the Fibonacci Sequence?
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. Starting with 0 and 1, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
Mathematical Definition
F₀ = 0, F₁ = 1
Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2
Binet's Formula
For direct calculation of the nth term:
Fₙ = (φⁿ - ψⁿ) / √5
where φ = (1+√5)/2 and ψ = (1-√5)/2
Applications in Nature
- 🌻Sunflower seed spirals (often 21, 34, 55, or 89)
- 🐚Nautilus shell spiral chambers
- 🌿Leaf arrangements and flower petals
- 🏗️Architecture and art proportions
Negative Fibonacci Numbers
The sequence extends to negative indices using:
F₋ₙ = Fₙ × (-1)ⁿ⁺¹
Example: F₋₅ = F₅ × (-1)⁶ = 5 × 1 = 5
Custom Sequences
Any sequence following the same recurrence relation Fₙ = Fₙ₋₁ + Fₙ₋₂ with different starting values creates a "generalized Fibonacci sequence."