Calculate Fibonacci Numbers

Online calculator and formulas for computing a Fibonacci number at a given index

Fibonacci Number Calculator

Fibonacci Sequence

The calculator computes the nth Fibonacci number from the famous number sequence, where each number is the sum of the two preceding ones.

Whole number ≥ 0 for the position in the Fibonacci sequence
Result
F(n):

Fibonacci Sequence

First 15 Fibonacci Numbers:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377
Golden Ratio

The ratio of consecutive Fibonacci numbers approaches the Golden Ratio φ ≈ 1.618.

\[\phi = \frac{1 + \sqrt{5}}{2} \approx 1{.}618034\]

The Fibonacci sequence appears frequently in nature: flower petals, pine cones, nautilus shells, and galaxy spirals.

Fibonacci Sequence Formulas

Fibonacci Formula
Binet's Formula
\[F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}\]

Where φ = (1+√5)/2 and ψ = (1-√5)/2

Recursive Definition
\[F_n = F_{n-1} + F_{n-2}\]

With F₀ = 0 and F₁ = 1

Examples

Calculation Examples
  • F(0) = 0
  • F(1) = 1
  • F(6) = 8
  • F(10) = 55
  • F(20) = 6765
Applications

Fibonacci numbers are used in computer science, stock market analysis, architecture, and art.

Detailed Description of Fibonacci Numbers

Mathematical Definition

The Fibonacci sequence is an infinite sequence of numbers that begins with 0 and 1. Each subsequent number is the sum of the two preceding numbers. This sequence was described in 1202 by Leonardo Fibonacci in his work "Liber Abaci".

Using the Calculator

Enter the index (starting with 0) and click 'Calculate'. The calculator uses Binet's formula for efficient computation.

Properties
  • Growth: Exponential growth with base φ
  • Ratios: Fₙ₊₁/Fₙ → φ (Golden Ratio)
  • Identities: Cassini identity, Catalan identity

Applications and Occurrences

In Nature
  • Flower petals (lilies: 3, buttercups: 5)
  • Pine cones and pineapples (spirals)
  • Sunflower seeds (spiral patterns)
  • Nautilus shells
In Technology
  • Algorithms and data structures
  • Fibonacci heap in computer science
  • Optimization and search algorithms
  • Pseudo-random number generation
In Art and Architecture
  • Golden ratio in paintings
  • Architectural proportions
  • Music composition and harmony
  • Photography (rule of thirds)
Interesting Facts
  • Every 3rd Fibonacci number is even
  • Sum of first n Fibonacci numbers: Fₙ₊₂ - 1
  • The rabbit problem led to its discovery
  • Related to the Golden Ratio

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