Enter a value for n to find the nth Fibonacci number, or generate a Fibonacci sequence between two positions. Choose between Single Number mode to get one specific term, or Sequence mode to list all Fibonacci numbers from start n to end n. Results include the calculated value(s) plus a visual breakdown of the sequence.
Fibonacci Number F(n)
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Sequence Length
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F(n-1)
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F(n+1)
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The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It starts with F(0) = 0 and F(1) = 1, so the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The rule is simply F(n) = F(n-1) + F(n-2).
You can calculate any Fibonacci number directly using Binet's formula: F(n) = ((1 + √5)^n − (1 − √5)^n) / (2^n × √5). This closed-form expression lets you find any term without computing all previous terms, though for very large n, iterative methods are more numerically stable.
By modern convention, the sequence starts at F(0) = 0, followed by F(1) = 1 and F(2) = 1. Some older definitions begin at F(1) = 1, F(2) = 1, skipping the zero term. This calculator uses the F(0) = 0 convention.
This calculator supports values of n from 0 up to 500. Fibonacci numbers grow exponentially, so by n = 500 the value has over 100 digits. Results for very large n are displayed using JavaScript's BigInt arithmetic to maintain full precision.
As n increases, the ratio of consecutive Fibonacci numbers F(n)/F(n-1) converges to the Golden Ratio φ ≈ 1.6180339887. This irrational number appears throughout mathematics, art, and nature, and is directly encoded in Binet's Fibonacci formula.
Yes — the sequence can be extended to negative indices using the rule F(-n) = (-1)^(n+1) × F(n). For example, F(-1) = 1, F(-2) = -1, F(-3) = 2. This calculator focuses on non-negative n (0 to 500).
Fibonacci numbers appear in many natural patterns — the spirals of sunflower seeds, pinecones, and galaxies often follow Fibonacci counts. They are also used in computer science algorithms, financial technical analysis (Fibonacci retracements), and art composition (Golden Rectangle).
In technical analysis, Fibonacci retracement levels (23.6%, 38.2%, 50%, 61.8%, 78.6%) are horizontal lines drawn between a high and low price to identify potential support and resistance levels. These ratios are derived from the relationships between numbers in the Fibonacci sequence.