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Fibonacci Sequence Calculator Generate the Fibonacci sequence up to N terms with exact BigInt precision.

Fibonacci Calculator illustration
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Fibonacci Calculator

Generate the Fibonacci sequence up to N terms with exact BigInt precision.

1

Enter Number of Terms

Specify how many Fibonacci numbers you want to generate (up to 500).

2

View the Sequence

See each Fibonacci number labeled with its index F(0), F(1), F(2), and so on.

3

Check Statistics

View the last term's value and its digit count.

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What Is Fibonacci Calculator?

The Fibonacci sequence is one of the most famous sequences in mathematics, defined by the recurrence relation F(n) = F(n−1) + F(n−2), with starting values F(0) = 0 and F(1) = 1. This produces the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The ratio of consecutive Fibonacci numbers converges to the golden ratio φ = (1+√5)/2 ≈ 1.618, which appears throughout nature in spiral patterns of sunflowers, pine cones, and galaxy arms. Fibonacci numbers also appear in computer science (in the analysis of the Euclidean algorithm and Fibonacci heaps), financial markets (Fibonacci retracement levels), and combinatorics (counting compositions). This calculator generates exact Fibonacci values using BigInt arithmetic, supporting up to 500 terms with full precision even for numbers with hundreds of digits.

Why Use Fibonacci Calculator?

  • Generates exact Fibonacci numbers using BigInt precision
  • Supports up to 500 terms
  • Displays index labels for each term
  • Shows digit count for large Fibonacci numbers
  • Clean visual layout with scrollable results

Common Use Cases

Mathematics Education

Study the properties of the Fibonacci sequence and the golden ratio.

Algorithm Analysis

Analyze algorithms related to Fibonacci numbers, such as the Euclidean algorithm's worst case.

Nature & Art

Explore the mathematical patterns found in natural spirals and artistic compositions.

Financial Analysis

Reference Fibonacci levels used in technical analysis and trading strategies.

Technical Guide

The Fibonacci sequence is generated iteratively using JavaScript BigInt for exact arbitrary-precision integer arithmetic. The iterative approach (vs. naive recursion) runs in O(n) time and O(n) space, compared to O(2^n) for naive recursion. The sequence grows approximately exponentially with ratio φ ≈ 1.618: F(n) ≈ φⁿ/√5. This means the number of digits in F(n) is approximately n × log₁₀(φ) ≈ n × 0.209. At n = 500, F(500) has about 105 digits. Binet's formula gives an exact closed-form: F(n) = (φⁿ − ψⁿ)/√5 where ψ = (1−√5)/2 ≈ −0.618, but floating-point imprecision makes this unreliable for large n, which is why this calculator uses iterative BigInt computation instead. Notable Fibonacci properties include: every third number is even, GCD(F(m), F(n)) = F(GCD(m,n)), and the sum of the first n Fibonacci numbers equals F(n+2) − 1.

Tips & Best Practices

  • 1
    The ratio F(n+1)/F(n) approaches the golden ratio φ ≈ 1.618 as n increases
  • 2
    Every 3rd Fibonacci number is even, every 4th is divisible by 3, every 5th by 5
  • 3
    F(n) has approximately n × 0.209 digits
  • 4
    The worst case for the Euclidean algorithm involves consecutive Fibonacci numbers
  • 5
    Fibonacci numbers can tile rectangles and create natural-looking spirals

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Frequently Asked Questions

Q What is the golden ratio?
The golden ratio φ = (1+√5)/2 ≈ 1.61803398875. The ratio of consecutive Fibonacci numbers converges to this value as the numbers get larger.
Q Why start with 0?
The modern convention starts with F(0) = 0, F(1) = 1. Some older sources start with F(1) = 1, F(2) = 1. Both produce the same sequence, just with shifted indices.
Q How large can the numbers get?
F(500) has about 105 digits. The calculator uses BigInt arithmetic, so all digits are exact with no rounding.
Q Are Fibonacci numbers used in trading?
Yes, Fibonacci retracement levels (23.6%, 38.2%, 50%, 61.8%) are widely used in technical analysis to identify potential support and resistance levels.
Q What are some Fibonacci number properties?
Every third is even, GCD(F(m),F(n)) = F(GCD(m,n)), and the sum of first n Fibonacci numbers equals F(n+2) − 1.

About This Tool

Fibonacci Calculator is a free online tool by FreeToolkit.ai. All processing happens directly in your browser — your data never leaves your device. No registration or installation required.