Fibonacci Number Calculator

F(n) = F(n−1) + F(n−2). 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Inputs

Result

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How to use this calculator

  • Enter n.
  • Read F(n) + ratio + golden-ratio comparison.

About this calculator

Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Each term = sum of previous two. Defined by F(0)=0, F(1)=1, F(n)=F(n−1)+F(n−2). Ratio F(n)/F(n−1) → golden ratio Φ = (1+√5)/2 ≈ 1.618 as n → ∞. Closed form (Binet): F(n) = (Φⁿ − (−1/Φ)ⁿ) / √5. Appears in nature: pinecones, sunflowers, nautilus shells, branching patterns, art (Parthenon proportions).

Frequently asked

Where in nature?+
Sunflower seed spirals (34/55 commonly), pinecone scales, nautilus shell proportions, tree branching, leaf phyllotaxis.
Golden ratio?+
Φ = (1+√5)/2 ≈ 1.61803. Aesthetically pleasing proportion claimed in art (Parthenon, Mona Lisa).
Binet's formula?+
F(n) = (Φⁿ − ψⁿ)/√5 where ψ = (1−√5)/2. Exact closed-form.
Big numbers?+
F(80) ≈ 2.3 × 10¹⁶. F(100) overflows JS Number. Use BigInt for n > 80.
Why so famous?+
Self-similar growth pattern emerges everywhere. Plus elegant recursion + golden ratio link.

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