Fibonacci Sequence Generator & Nth Term Calculator
Generate the first N Fibonacci numbers or find the exact Nth term with BigInt precision, exact past F(78) where floating-point math starts rounding.
Other seeds give other sequences with the same rule — the Lucas numbers, for one.
0, 1, 1, 2, 3, 5, 8, 13, 21, 340, 1, 1, 2, 3, 5, 8, 13, 21, 34
The golden ratio
F(9) ÷ F(8) = 1.6190476190
φ = (1 + √5) / 2 = 1.6180339887…
Divide any term by the one before it and the answer gets closer to the golden ratio φ the further along the sequence you go — each term is roughly the last one times φ.
The spiral
Exact BigInt math keeps even huge terms precise.
The whole list, or just F(N)
Enter a value N and this tool instantly returns either the first N numbers of the Fibonacci sequence or the single Nth term, depending on the mode you pick. The Fibonacci sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21… where every number is the sum of the two before it. Indexing is zero-based, so F(0) = 0, F(1) = 1 and F(10) = 55. Everything updates live as you type, and a copy button lets you grab the result in one click.
Why exact big-number math matters
Fibonacci numbers grow exponentially — F(100) is already 21 digits long — so ordinary floating-point numbers silently lose precision and start returning rounded, wrong digits past F(78) or so. This generator uses JavaScript BigInt arithmetic, which stores integers of unlimited size, so every term is shown with all of its digits intact. To keep output readable and fast, N is capped at 1000, which still produces terms with more than 200 digits. In the Nth-term mode with the usual starting terms, “Properties of F(N)” adds how many digits the term has and, from the rule that F(d) divides F(n) whenever d divides n, the factors of F(N) that the divisors of N hand you — enough, for most composite N, to show that F(N) is not prime.
The story the sequence came from
The numbers are named after Leonardo of Pisa, called Fibonacci, who put them in his arithmetic book Liber Abaci in 1202 as a puzzle about rabbits. A pair of newborn rabbits is shut in a walled field. Each pair takes a month to grow up and from then on produces one new pair every month, and no rabbit ever dies. How many pairs are there after a year? Count month by month and each month's total is last month's — every pair still alive — plus the month before's, because exactly those pairs are old enough to breed: 1, 1, 2, 3, 5, 8 and on, each number the sum of the two before it. The rules are what make the answer so tidy: rabbits that never die, every litter exactly one male and one female, breeding every month without fail, and a field that never runs out of grass. It is an idealised puzzle, not biology — real rabbits die, have several young at a time and run short of food — and the sequence earned its place in mathematics, not in zoology. Indian scholars had met the same numbers centuries earlier, counting the rhythms of Sanskrit verse.
Frequently asked questions
Does the sequence start at 0 or 1?
This tool uses the standard zero-based convention: F(0) = 0, F(1) = 1, F(2) = 1, and so on. So the first N terms always begin with 0, and the Nth term counts from index 0.
Can it show very large Fibonacci numbers without rounding?
Yes. The generator uses BigInt integer math, so results are exact to the last digit no matter how many digits they have — unlike calculators that switch to scientific notation and round large terms.
Can I start from other numbers, or find where a value falls in the sequence?
Yes. The two starting terms can be changed, and “Up to a value” lists the terms up to a number and says which term it is.
Can N be negative, and where does the golden ratio come in?
Yes, in the Nth-term mode: the rule runs backwards, each term being the one two places after it minus the one right after it, and the working lists those subtractions; the first-N-terms mode still needs a count of 0 or more. Beside the answer, “The golden ratio” divides the last two terms on screen and sets φ = (1 + √5) / 2 next to it, and “The spiral” draws the terms as squares laid edge to edge with a quarter circle across each — for the usual seeds, the only ones that tile.
Can it tell me anything about F(N) besides its value?
Yes. In the Nth-term mode with the usual starting terms, “Properties of F(N)” gives the number of digits and uses the rule that F(d) divides F(n) whenever d divides n: every divisor of N other than 1, 2 and N itself hands over a factor of F(N), listed with its value, which shows that F(N) is not prime. When N has no such divisor — N prime, or 4 — the page says the rule finds no factor, and F(N) may or may not be prime.