Calculate the n-th Fibonacci number and explore this iconic mathematical sequence found throughout nature.
Last updated: May 2026 | By Patchworkr Team
| n | F(n) |
|---|---|
| F(0) | 0 |
| F(1) | 1 |
| F(5) | 5 |
| F(10) | 55 |
| F(15) | 610 |
| F(20) | 6765 |
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting with 0 and 1. It appears in nature: flower petals, spiral galaxies, tree branching, and pineapple fruitlets all exhibit Fibonacci patterns. The sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Mathematically, this sequence demonstrates rapid exponential growth and connects to the Golden Ratio: as n grows, the ratio F(n)/F(n-1) approaches φ ≈ 1.618. Applications span computer science (algorithm analysis), finance (technical analysis), physics, and art. Computing Fibonacci numbers efficiently demonstrates important concepts like recursion, dynamic programming, and algorithm complexity.
Base cases: F₀ = 0, F₁ = 1
Leonardo of Pisa (Fibonacci) introduced it to Western math in Liber Abaci (1202) through a rabbit problem.
Ratios of consecutive Fibonacci numbers approach φ ≈ 1.618 as n grows — the mathematically ideal proportion.
Flower petals (lilies have 3, buttercups 5), tree branching, spiral galaxies, and many organic structures follow Fibonacci patterns.
Demonstrating recursion, dynamic programming, algorithm complexity, and data structure design (Fibonacci heaps).
A closed-form: $F_n = \frac{φ^n - ψ^n}{\sqrt{5}}$ — useful for analysis but numerically unstable for large n.
Use matrix exponentiation or fast-doubling algorithms with big integers for efficiency and precision.
Yes: $F_{-n} = (-1)^{n+1} F_n$ — the sequence extends infinitely in both directions.
Financial forecasting (technical analysis), search algorithms, network optimization, and digital signal processing.
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