The Fibonacci sequence starts 0, 1 and then adds the last two terms to get the next: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … The numbers grow by a factor approaching the golden ratio, about 1.618, at every step, so the 100th term already has 21 digits. This calculator finds any term up to F(100,000) exactly, lists up to 500 consecutive terms, and tests whether a number of up to 10,000 digits is a Fibonacci number. It also handles Lucas numbers and sequences with any two starting values.
How to use the Fibonacci calculator
- Choose a mode: Find the nth term, List terms or Is it Fibonacci?
- For a single term, enter Term number (n). Terms are counted from F(0) = 0.
- For a list, enter First term to list and How many terms (up to 500). The list appears as a table with the ratio between neighbors, plus a comma-separated copy you can paste elsewhere.
- Under Sequence, keep the standard Fibonacci start, switch to Lucas numbers, or choose custom first two terms.
- To test a number, type it in Number to test. Commas and spaces are ignored.
Fibonacci formulas
The defining rule, or recurrence:
Binet’s closed form, with φ = (1 + √5)/2 and ψ = (1 − √5)/2:
Binet’s formula is perfect in theory, but a computer using ordinary decimals runs out of precision around F(70). To keep every digit exact, the calculator uses whole-number arithmetic and the doubling identities F(2k) = F(k)(2F(k + 1) − F(k)) and F(2k + 1) = F(k)² + F(k + 1)², which reach F(100,000) in about 17 steps.
Worked example
Find F(12). Build up from the start: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
F(12) = F(11) + F(10) = 89 + 55 = 144.
Binet check: φ12 ÷ √5 ≈ 144.0014, which rounds to 144.
Test in reverse: 5 × 144² + 4 = 103,684 = 322², a perfect square, so 144 is a Fibonacci number.
Fibonacci numbers worth knowing
| n | F(n) | n | F(n) |
|---|---|---|---|
| 10 | 55 | 50 | 12,586,269,025 |
| 12 | 144 | 60 | 1,548,008,755,920 |
| 20 | 6,765 | 80 | 23,416,728,348,467,685 |
| 30 | 832,040 | 100 | 354,224,848,179,261,915,075 |
| 40 | 102,334,155 | 1,000 | 209 digits |
Properties and patterns
- Golden ratio. F(n + 1) ÷ F(n) approaches φ ≈ 1.6180339887 — the list mode shows the ratio settling down. This works for any starting pair of positive numbers; see the golden ratio calculator.
- Sums. F(0) + F(1) + ⋯ + F(n) = F(n + 2) − 1. The first eleven terms, up to 55, add to 143 = 144 − 1.
- Even terms. Every third Fibonacci number is even: F(3) = 2, F(6) = 8, F(9) = 34.
- Divisibility. The greatest common divisor of F(m) and F(n) is F(gcd(m, n)). So F(n) divides F(kn), and consecutive Fibonacci numbers share no common factor.
- Pascal’s triangle. Adding the numbers along the shallow diagonals of Pascal’s triangle gives 1, 1, 2, 3, 5, 8, …
- Zeckendorf’s theorem. Every positive whole number is a sum of non-consecutive Fibonacci numbers in exactly one way, for example 100 = 89 + 8 + 3.
Where the sequence comes from
The sequence is named after Leonardo of Pisa, known as Fibonacci, whose 1202 book Liber Abaci posed a puzzle about a population of rabbits that doubles in a particular way each month. The same numbers had appeared centuries earlier in Indian work on poetic meter, where they count the rhythms that can be made from short and long syllables. The closed formula is credited to Jacques Binet, who published it in 1843, though Abraham de Moivre knew it more than a century before. Édouard Lucas studied the sequence closely in the 1870s and gave it the name Fibonacci; the companion sequence 2, 1, 3, 4, 7, 11, … carries his own name.
Frequently asked questions
What is the Fibonacci sequence?
It is the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, … in which each term is the sum of the two before it. In symbols, F(0) = 0, F(1) = 1 and F(n) = F(n − 1) + F(n − 2).
What is the 100th Fibonacci number?
Counting from F(0) = 0, F(100) = 354,224,848,179,261,915,075, a 21-digit number. Some books start the sequence at F(1) = 1, F(2) = 1, which gives the same values with the same index.
Is there a formula for the nth Fibonacci number?
Yes, Binet's formula: F(n) = (φⁿ − ψⁿ) ÷ √5, where φ = (1 + √5)/2 ≈ 1.618 and ψ = (1 − √5)/2 ≈ −0.618. Because the ψ term is tiny, F(n) is simply φⁿ ÷ √5 rounded to the nearest whole number.
How can I tell whether a number is a Fibonacci number?
A whole number x is a Fibonacci number exactly when 5x² + 4 or 5x² − 4 is a perfect square. For 144, 5 × 144² + 4 = 103,684 = 322², so 144 is Fibonacci; it is F(12).
What are Lucas numbers?
Lucas numbers follow the same rule but start with 2 and 1: 2, 1, 3, 4, 7, 11, 18, 29, … They are named after the French mathematician Édouard Lucas, and L(n) = F(n − 1) + F(n + 1).