Fibonacci Sequence Calculator

F(n) = F(n−1) + F(n−2)

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About Fibonacci Sequence Calculator

A Fibonacci calculator computing F(n) using matrix exponentiation and Binet's formula. Shows the golden ratio φ, Lucas numbers, and sequence terms. Verifies identities like Cassini's. All calculations are client-side.

Fibonacci Sequence Calculator Features

  • F(n)
  • Binet formula
  • Golden ratio
  • Sequence
  • Identities
Fibonacci: F(0)=0, F(1)=1, F(n)=F(n−1)+F(n−2). Binet: F(n) = (φⁿ−ψⁿ)/√5 where φ=(1+√5)/2, ψ=(1−√5)/2. F(n)/F(n−1) → φ. Growth: F(n) ∼ φⁿ/√5.

How to Use

Enter n:

  • n: Index
  • F(n): Fibonacci number
  • Sequence: First n+1 terms

Identities

  • Cassini: F(n−1)F(n+1)−F(n)² = (−1)ⁿ
  • Sum: ΣF(k) = F(n+2)−1
  • gcd(F(m),F(n)) = F(gcd(m,n))

Applications

  • Nature: phyllotaxis, spirals
  • Art: golden ratio proportions
  • CS: Fibonacci heaps, search
  • Finance: Fibonacci retracements

Step-by-Step Instructions

  1. 1Enter n.
  2. 2Get F(n).
  3. 3View sequence.
  4. 4Check Binet.
  5. 5Verify identities.

Fibonacci Sequence Calculator — Frequently Asked Questions

What is the golden ratio?+

φ = (1+√5)/2 ≈ 1.6180339887. It satisfies φ² = φ+1. F(n)/F(n-1) → φ. A rectangle with ratio φ has the property that cutting a square leaves a similar rectangle.

How is Fibonacci computed efficiently?+

Matrix method: [F(n+1),F(n); F(n),F(n-1)] = [1,1;1,0]ⁿ. Matrix exponentiation gives O(log n) time. Binet's formula works in O(1) but has precision issues for large n.

Where do Fibonacci numbers appear in nature?+

Sunflower spirals (21/34 or 34/55), pinecone scales, pineapple eyes, leaf arrangements — always consecutive Fibonacci numbers. This maximizes packing efficiency via the golden angle 360°/φ² ≈ 137.5°.

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