How is the Fibonacci sequence defined?

Each term is the sum of the two before it: F(n) = F(n-1) + F(n-2). This reference starts the sequence at F(1) = 1 and F(2) = 1, so the list runs 1, 1, 2, 3, 5, 8, 13, and onward.

Does the indexing start at 0 or 1?

This tool is 1-indexed: F(1) = 1 and F(2) = 1. Some textbooks use F(0) = 0 instead. If you need the 0-indexed value, subtract one from the index you enter here.

How large can the numbers get without rounding errors?

All values use JavaScript BigInt, so they stay exact no matter how many digits long. Regular floating-point numbers would lose precision past F(78); BigInt avoids that entirely.

What is the connection to the golden ratio?

The ratio of consecutive Fibonacci numbers F(n+1)/F(n) converges to the golden ratio φ ≈ 1.6180339887 as n grows. That relationship underlies many of the sequence's appearances in geometry and nature.

Where are Fibonacci numbers used in programming?

They appear in algorithm analysis, dynamic-programming examples, Fibonacci heaps, and as a classic recursion or memoization benchmark. Quick lookups help when testing or hand-checking such code.

Fibonacci Sequence Reference

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The Fibonacci sequence is one of the most famous integer sequences in mathematics: every term equals the sum of the two preceding terms. This reference lists the numbers from F(1) onward and lets you jump straight to any single F(n) — computed exactly, with no rounding, directly in your browser.

How it works

The sequence is built from a simple recurrence with two seed values:

F(1) = 1
F(2) = 1
F(n) = F(n-1) + F(n-2)   for n > 2

To produce the list, the tool keeps a running pair of the last two values and adds them to generate the next, repeating up to the count you choose. The single-index lookup runs the same loop up to your chosen n and returns the final term. Because each addition uses BigInt, the values never overflow or lose precision — F(100), for example, is the exact 21-digit integer 354224848179261915075.

Tips and example

A 1-indexed list begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 — handy for hand-checking code that generates the sequence.
The ratio of neighbouring terms approaches the golden ratio φ ≈ 1.618; by F(20) it is already accurate to several decimals.
If your project defines F(0) = 0, just remember this tool starts at F(1); shift your index by one to match.

Use the copy button to grab an exact large Fibonacci number for a test fixture or a constant, without re-deriving it by hand.