Home › Calculators › Fibonacci Calculator

Fibonacci Calculator

Find any Fibonacci number exactly, even with thousands of digits, list the sequence from any starting point, test whether a number is a Fibonacci number, and watch the ratio of consecutive terms converge to the golden ratio φ.

What do you want?

n-th Fibonacci numberList of Fibonacci numbersIs it a Fibonacci number?Golden ratio convergence

n (position)

F(0) = 0, F(1) = 1. Exact up to n = 100,000; negative n gives negafibonacci numbers

How many terms

Up to 1,000 terms

Start at n =

Number to check

Whole numbers of any size

Ratios up to n =

Enter your values to see the result.

How to use the Fibonacci Calculator

Choose a mode: n-th number, list, check, or golden ratio.
Enter n, the number of terms, or the number to check.
Press Calculate.
Read the result; for very large numbers the full value is shown below.
Copy the results or download the list as CSV.

Fibonacci formulas

Formula	Meaning
F(n) = F(n − 1) + F(n − 2), F(0) = 0, F(1) = 1	Recursive definition
F(n) = (φⁿ − ψⁿ)/√5	Binet's formula, φ = (1 + √5)/2, ψ = (1 − √5)/2
F(2k) = F(k)(2F(k + 1) − F(k))	Fast doubling (even index)
F(2k + 1) = F(k)² + F(k + 1)²	Fast doubling (odd index)
F(0) + F(1) + … + F(n) = F(n + 2) − 1	Sum of the first terms
F(n − 1)·F(n + 1) − F(n)² = (−1)ⁿ	Cassini's identity
L(n) = F(n − 1) + F(n + 1)	Lucas numbers (2, 1, 3, 4, 7, 11, …)

The first Fibonacci numbers

n	F(n)	n	F(n)	n	F(n)
0	0	10	55	20	6,765
1	1	11	89	21	10,946
2	1	12	144	22	17,711
3	2	13	233	23	28,657
4	3	14	377	24	46,368
5	5	15	610	25	75,025
6	8	16	987	30	832,040
7	13	17	1,597	40	102,334,155
8	21	18	2,584	50	12,586,269,025
9	34	19	4,181	100	354,224,848,179,261,915,075

Fun properties

Every 3rd Fibonacci number is even, every 4th is divisible by 3 and every 5th by 5.
F(m) divides F(n) whenever m divides n, and gcd(F(m), F(n)) = F(gcd(m, n)).
F(n) has roughly n × 0.209 digits, since log₁₀ φ ≈ 0.20899.
Fibonacci numbers appear in sunflower seed spirals, pine cones and the branching of plants.

Frequently asked questions

What is the Fibonacci sequence?

A sequence that starts 0, 1 and where each next number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … In symbols, F(0) = 0, F(1) = 1 and F(n) = F(n − 1) + F(n − 2).

What is the 100th Fibonacci number?

With F(0) = 0 and F(1) = 1, F(100) = 354,224,848,179,261,915,075, a 21-digit number. F(50) = 12,586,269,025.

How can I tell if a number is a Fibonacci number?

A positive whole number n is a Fibonacci number exactly when 5n² + 4 or 5n² − 4 is a perfect square. For 144: 5 × 144² − 4 = 103,676 = 322², so 144 is a Fibonacci number (it is F(12)).

How is the Fibonacci sequence related to the golden ratio?

The ratio of consecutive terms F(n + 1)/F(n) approaches φ = (1 + √5)/2 ≈ 1.6180339887. Binet's formula F(n) = (φⁿ − ψⁿ)/√5, with ψ = (1 − √5)/2, makes this exact; F(n) is the whole number nearest to φⁿ/√5.

Why can this calculator show such large numbers exactly?

It uses JavaScript BigInt, which stores integers with any number of digits, and the fast doubling method F(2k) = F(k)(2F(k + 1) − F(k)), F(2k + 1) = F(k)² + F(k + 1)², which needs only about log₂(n) steps.

Does the sequence start with 0 or 1?

Modern mathematics uses F(0) = 0 and F(1) = 1, which this calculator follows. Some books start with F(1) = F(2) = 1; the numbers are the same, only the index labels differ.

What are negafibonacci numbers?

Running the rule backwards gives Fibonacci numbers for negative n: F(−n) = (−1)^(n + 1) F(n). So F(−1) = 1, F(−2) = −1, F(−3) = 2, F(−4) = −3.

Related tools