What is the Fibonacci sequence?

The Fibonacci sequence is the infinite series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... where each term is the sum of the two before it. Formally F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for all n greater than or equal to 2. It appears in phyllotaxis (leaf and seed arrangements), spiral galaxies, financial models, and computer-science algorithms.

How is F(n) computed for large n?

The calculator uses the fast-doubling algorithm, which runs in O(log n) steps using only two BigInt variables. The key identities are: F(2k) = F(k) * (2*F(k+1) - F(k)) and F(2k+1) = F(k)^2 + F(k+1)^2. This lets the calculator reach F(10000) — a 2,090-digit integer — in milliseconds without storing the whole sequence.

What is Binet's closed-form formula?

Binet's formula states F(n) = round( phi^n / sqrt(5) ) where phi = (1 + sqrt(5)) / 2 is approximately 1.6180339887. It is exact for all non-negative integers but loses precision in floating-point arithmetic beyond about n = 78 because phi^n overflows a 64-bit double. The calculator shows the closed-form result for small n and switches to exact BigInt fast-doubling for larger values.

What is the golden ratio and how does Fibonacci relate to it?

The golden ratio phi = (1 + sqrt(5)) / 2 is approximately 1.61803. As n grows, the ratio F(n) / F(n-1) converges to phi. By n = 10 the ratio is already 1.6179..., within 0.01% of phi. This is why Fibonacci numbers appear wherever the golden ratio shows up in geometry, art, and nature. Use the "Golden-ratio convergence" view to watch the error shrink with each term.

How many digits does F(1000) have?

F(1000) has 209 decimal digits. F(10000) has 2,090 digits. The number of digits in F(n) grows as approximately n * log10(phi), which is about 0.209 * n. The calculator computes and displays the full exact integer for every value up to n = 10,000.

Is the first Fibonacci number 0 or 1?

There are two common conventions. This calculator uses the modern zero-indexed convention: F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, ... If you need the one-indexed version (starting 1, 1, 2, 3, ...) simply add 1 to your index — so the 10th term under that convention equals F(10) = 55 in this calculator.

Fibonacci Calculator

The Fibonacci sequence is one of the most recognisable patterns in mathematics. Starting from 0 and 1, every subsequent number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … The sequence turns up in sunflower seed spirals, nautilus shells, tree branching patterns, financial Elliott-wave analysis, and the analysis of algorithms. This calculator lets you find any term instantly, copy the full exact integer (even when it runs to hundreds of digits), explore how the ratio of consecutive terms converges to the golden ratio phi, and see the classic rectangle spiral drawn as an SVG.

How it works

The calculator uses the fast-doubling algorithm, a divide-and-conquer method that computes F(n) in O(log n) arithmetic operations using two variables and JavaScript’s built-in BigInt type for perfect precision. It relies on the identities:

F(2k)   = F(k) * (2*F(k+1) - F(k))
F(2k+1) = F(k)^2 + F(k+1)^2

Starting from F(0) = 0 and F(1) = 1, the algorithm reads the binary representation of n from the most-significant bit down, halving the problem at each step. This is far faster than the naive recurrence loop for large n — reaching F(10,000) in milliseconds where a loop would need ten thousand additions of increasingly large BigInts.

For small n (up to 78) the tool also shows the Binet / closed-form approximation:

F(n) = round( phi^n / sqrt(5) )

where phi = (1 + sqrt(5)) / 2 is the golden ratio. Beyond n = 78 floating-point precision is exhausted, so only the exact BigInt result is shown.

Worked example

Find F(10):

Start: F(0) = 0, F(1) = 1
Apply the recurrence eight more times: F(2) = 1, F(3) = 2, F(4) = 3, F(5) = 5, F(6) = 8, F(7) = 13, F(8) = 21, F(9) = 34
F(10) = F(9) + F(8) = 34 + 21 = 55

Closed-form check: phi^10 / sqrt(5) = 1.6180^10 / 2.2360 = 122.99 / 2.236 = 55.00. Rounds to 55. Exact.

n	F(n)	Ratio F(n)/F(n-1)	Digits
5	5	1.6667	1
10	55	1.6182	2
20	6,765	1.61804	4
50	12,586,269,025	1.6180339887	11
100	354,224,848,179,261,915,075	1.61803398875	21
1000	(209-digit integer)	phi to 15 decimal places	209

Formula note

The general term can also be expressed using the conjugate root psi = (1 - sqrt(5)) / 2 as the exact closed form:

F(n) = ( phi^n - psi^n ) / sqrt(5)

Because |psi| is less than 1 (approximately -0.618), the term psi^n becomes negligibly small for large n and the formula reduces to the rounding version above. This is Binet’s formula, published in 1843, though Abraham de Moivre knew an equivalent form a century earlier.

Every calculation runs entirely in your browser using native BigInt arithmetic. No numbers are sent to a server or stored anywhere.