Fibonacci Calculator
Generate the Fibonacci sequence up to any term and check if a number belongs to the sequence.
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How to use this calculator
Each Fibonacci number is the sum of the two preceding numbers, starting from 1, 1.
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Enter N to generate the first N Fibonacci numbers (up to 30).
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Enter a number in the check field to see if it belongs to the Fibonacci sequence.
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Watch the ratio F(n)/F(n−1) approach the golden ratio φ ≈ 1.618.
Frequently asked questions
What is the golden ratio?
The golden ratio φ = (1 + √5) / 2 ≈ 1.618. The ratio of consecutive Fibonacci numbers (F(n+1)/F(n)) converges to φ as n grows. The golden ratio appears in art (Parthenon proportions), nature (spiral shells, sunflower seed arrangements), and has the remarkable property that φ² = φ + 1.
Where does the Fibonacci sequence appear in nature?
The number of spiral rows in sunflower seed heads, pinecone scales, and pineapple bracts are typically consecutive Fibonacci numbers (8/13, 13/21, 21/34). Leaf arrangements (phyllotaxis) on stems follow Fibonacci numbers to maximise light exposure. Romanesco broccoli exhibits fractal Fibonacci spirals.
What is Binet's formula?
F(n) = (φⁿ − ψⁿ) / √5, where φ = (1+√5)/2 and ψ = (1−√5)/2. This closed-form formula computes any Fibonacci number directly without iteration. Since |ψ| < 1, for large n: F(n) ≈ φⁿ / √5, rounded to the nearest integer.
What is the Fibonacci sequence used for in algorithms?
Fibonacci heaps (a data structure), Fibonacci search (faster than binary search in some cases), recursive algorithm analysis (naïve recursive Fibonacci demonstrates exponential time complexity), and dynamic programming examples. Fibonacci numbers also define the worst case for Euclid's GCD algorithm.
The Fibonacci sequence in mathematics and nature
History of the Fibonacci sequence
Leonardo of Pisa (nicknamed Fibonacci) introduced the sequence to Western mathematics in 1202 in "Liber Abaci" through a rabbit breeding problem. However, the sequence was described centuries earlier in Indian mathematics by Virahanka (circa 700 AD) in the context of poetic meter.
Fibonacci numbers in financial markets
Fibonacci retracement levels (23.6%, 38.2%, 61.8%, 78.6%) are widely used in technical analysis. Traders draw horizontal lines at these levels between a swing high and low to identify potential support and resistance zones. While their predictive power is debated, their widespread use makes them self-fulfilling to some degree.
Fibonacci and the golden spiral
Drawing quarter-circles in consecutive Fibonacci-sized squares creates the Fibonacci spiral, which approximates the golden spiral (based on φ). This spiral appears in nautilus shells, hurricane bands, and galaxy arm shapes — though the match is often approximate rather than exact.
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