MathCalcLab

Fibonacci Sequence Calculator

Calculate Fibonacci numbers with golden ratio connection and sequence visualization

Enter a non-negative integer (0 to 10000)

Résultats

Entrez les valeurs et cliquez sur Calculer pour voir le résultat.

Theory & Formula

Theory

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. Named after Leonardo Fibonacci (13th century), this sequence appears extensively in nature, art, and mathematics.

Recursive Formula

\(F_n = F_{n-1} + F_{n-2}\)\(F_0 = 0, \quad F_1 = 1\)

Binet's Formula (Closed Form)

\(F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}\)

where \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\) (golden ratio)

Properties

  • Every 3rd Fibonacci number is divisible by 2
  • Every 4th Fibonacci number is divisible by 3
  • The ratio of consecutive Fibonacci numbers approaches φ (golden ratio)
  • Sum of first n Fibonacci numbers: F₁ + F₂ + ... + Fₙ = Fₙ₊₂ - 1

Applications

Fibonacci numbers appear in nature (spiral patterns in shells, flower petals, pine cones), computer science (algorithm analysis, dynamic programming), financial markets (Fibonacci retracement), and art & architecture (proportions).

Example

\(F_0 = 0\)\(F_1 = 1\)\(F_2 = F_1 + F_0 = 1 + 0 = 1\)\(F_3 = F_2 + F_1 = 1 + 1 = 2\)\(F_4 = F_3 + F_2 = 2 + 1 = 3\)\(F_5 = F_4 + F_3 = 3 + 2 = 5\)

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

Calculatrices associées

Combinations & Permutations

Calculate combinations and permutations

Sequences & Series Calculator

Arithmetic and geometric sequences

Factorial Calculator

Calculate factorials

Fibonacci Calculator | MathCalcLab | MathCalcLab