Fibonacci Sequence Calculator
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Understanding Fibonacci Sequences
Basic Concepts
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1.
Fₙ = Fₙ₋₁ + Fₙ₋₂
Initial Terms: F₀ = 0, F₁ = 1
Golden Ratio (φ) = (1 + √5)/2 ≈ 1.618034
Binet's Formula: Fₙ = (φⁿ - (-φ)⁻ⁿ)/√5
Properties and Characteristics
Key Properties
- Recursive nature
- Golden ratio convergence
- Exponential growth
- Divisibility patterns
- Lucas numbers relation
- Matrix representation
Special Relationships
- Golden spiral connection
- Pascal's triangle patterns
- Phyllotaxis in nature
- Cassini's identity
- Zeckendorf representation
- Pisano periods
Advanced Topics
Mathematical Properties
- Generating functions
- Continued fraction representation
- Fibonacci polynomials
- Q-matrix properties
- Fibonacci primes
- Negafibonacci numbers
Applications
Golden Rectangle: a/b = b/(a-b) = φ
Fibonacci Spiral: r = φᶿ/√5
Growth Factor: Fₙ₊₁/Fₙ → φ as n → ∞
Sum Formula: ΣFᵢ = Fₙ₊₂ - 1
Real-World Applications
- Financial market analysis
- Computer algorithms
- Architecture and design
- Natural growth patterns
- Music composition
- Stock trading
Extended Concepts
Tribonacci Numbers: Tₙ = Tₙ₋₁ + Tₙ₋₂ + Tₙ₋₃
Lucas Numbers: Lₙ = Lₙ₋₁ + Lₙ₋₂
Fibonacci Words: Binary sequence patterns