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Fibonacci Generator online
Generate the Fibonacci sequence — first N terms or enter a Fibonacci number
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The Fibonacci sequence starts with 0 and 1 (or 1 and 1), and each subsequent term is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… The sequence appears throughout nature (phyllotaxis, shell spirals, tree branching), computer science (search algorithms, dynamic programming), and financial technical analysis. This generator computes exact integer values using BigInt for arbitrarily large terms.
The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.618033… as n increases. Every third Fibonacci number is even (divisible by 2). Every fourth is divisible by 3. The sum of the first n Fibonacci numbers equals F(n+2) − 1. Fibonacci numbers that are also prime (Fibonacci primes) are relatively rare: 2, 3, 5, 13, 89, 233… Fibonacci's Pisano period gives the repeating pattern of remainders modulo any integer.
The golden ratio φ = (1 + √5) / 2 ≈ 1.6180339887. It is the limit of the ratio F(n+1)/F(n) as n approaches infinity, and appears in art, architecture, and nature. A rectangle with sides in the golden ratio is considered aesthetically pleasing.
Yes — Binet's formula: F(n) = (φⁿ − ψⁿ) / √5 where ψ = (1 − √5)/2. For exact integer results, the formula requires arbitrary-precision arithmetic because of the irrational numbers involved.
The number of spirals in sunflower seed heads, pineapple scales, and pine cones are typically consecutive Fibonacci numbers (e.g. 8 and 13, or 21 and 34). The angle between successive leaves on many plants approximates 137.5° — the golden angle derived from φ.
Fibonacci search is a divide-and-conquer search algorithm for sorted arrays that uses Fibonacci numbers to split the search space — similar to binary search but divides the array into golden ratio proportions rather than halves.
See also the Factorial Calculator, GCD/LCM Calculator, and the Bitwise Calculator.