Fibonacci Series Generator
Generate beautiful Fibonacci sequences and explore the mathematical wonders of this famous number pattern
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Your Fibonacci Sequence
Your generated sequence will appear here.
Sequence Statistics
Sequence Length
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Sum of Sequence
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Last Term
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Golden Ratio
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π Mathematical Properties of Fibonacci Sequence
Golden Ratio
The ratio of consecutive Fibonacci numbers approaches the golden ratio (β1.618) as the sequence progresses.
Sum Formula
The sum of the first n Fibonacci numbers is Fn+2 - 1.
Binet's Formula
Fn = (Οn - Οn)/β5, where Ο is golden ratio and Ο = 1 - Ο.
Divisibility
Every 3rd Fibonacci number is even, every 4th is divisible by 3, every 5th by 5, etc.
Infinite Sequence
The Fibonacci sequence continues infinitely in both positive and negative directions.
Nature's Pattern
Appears in flower petals, pinecones, hurricanes, galaxies, and many natural phenomena.
π‘ Real-World Applications
Computer Science
Used in algorithms, data structures (Fibonacci heaps), and efficient searching techniques.
Financial Markets
Fibonacci retracements are popular technical analysis tools in stock trading.
Architecture & Design
The golden ratio derived from Fibonacci numbers creates aesthetically pleasing proportions.
Biology
Models population growth of rabbits and appears in phyllotaxis (leaf arrangements).
π Famous Fibonacci Sequences
| Name | First Terms | Description |
|---|---|---|
| Classic Fibonacci | 0, 1, 1, 2, 3, 5... | The original sequence described by Fibonacci in 1202 |
| Lucas Numbers | 2, 1, 3, 4, 7, 11... | Similar sequence with different starting values |
| Tribonacci | 0, 0, 1, 1, 2, 4... | Each term is the sum of the previous three |
| Pell Numbers | 0, 1, 2, 5, 12, 29... | F(n) = 2*F(n-1) + F(n-2) |
| Negative Fibonacci | ..., -8, 5, -3, 2, -1, 1, 0, 1, 1, 2... | Sequence extended to negative indices |
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Note: This tool generates Fibonacci sequences for educational and mathematical purposes. For extremely large sequences, browser performance may be affected.