Fibonacci Sequence Generator

The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding terms. Starting from 0 and 1, the sequence begins:

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

Formula

The recurrence relation that defines the Fibonacci sequence:

F(n) = F(n-1) + F(n-2), where F(0) = 0, F(1) = 1

The Golden Ratio

As the sequence progresses, the ratio of consecutive Fibonacci numbers converges to the golden ratio (phi):

phi = (1 + sqrt(5)) / 2 = 1.6180339887...

This ratio appears throughout nature, art, and architecture — from the spiral of a nautilus shell to the proportions of the Parthenon.

Applications

• Computer Science — Algorithm analysis, dynamic programming, data structures (Fibonacci heaps)
• Mathematics — Number theory, combinatorics, linear algebra
• Nature — Leaf arrangements, flower petals, spiral shells, branching patterns
• Finance — Fibonacci retracement levels in technical analysis
• Art & Design — Compositional proportions based on the golden ratio

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