Fibonacci Generator: Sequence, n-th Term and Lucas
List the first n Fibonacci numbers or find the n-th term exactly, up to 1000 terms. Use custom seeds such as Lucas numbers, pick the start index, and compare F(n+1)/F(n) with the golden ratio.
About the Fibonacci Generator
The Fibonacci Generator lists the first n Fibonacci numbers or finds a single n-th term, using exact whole-number arithmetic, so even the 1000th term prints every one of its 209 digits with no rounding. In the Fibonacci sequence each number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on.
Change the two seeds to build other sequences with the same rule. Seeds 2 and 1 give the Lucas numbers, and seeds 1 and 1 give the sequence that starts at 1, 1. Choose whether the first seed is labelled F(0) or F(1), pick a separator (comma, space or new line), then copy the list or download it as a .txt file.
How to use it
Steps
- Choose First n terms to list a sequence or n-th term to find one number.
- Enter the count or term number, and change the two seeds if you want something other than 0 and 1.
- Pick the index of the first seed, then press Generate or Find term. Use Copy or Download .txt for a list.
Limits
A list holds 1 to 1000 terms and the n-th term can be up to 1000 positions after the first seed. Seeds are whole numbers of zero or more, with at most 40 digits each. Anything else shows a message under the form.
The rule and the golden ratio
Each term is built from the two before it:
The page shows the ratio of the last two terms next to the golden ratio. It settles quickly: 21/13 is 1.615..., 89/55 is 1.618181... and F(31)/F(30) = 1,346,269 / 832,040 matches the golden ratio to the first ten decimals. The ratio is rounded half up to 15 decimals, and it is hidden when the earlier term is 0 because dividing by zero has no answer. Any pair of positive seeds approaches the same limit, so Lucas numbers do too.
Worked examples
- First 10 terms from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
- Lucas numbers, 8 terms (seeds 2 and 1): 2, 1, 3, 4, 7, 11, 18, 29.
- F(10) is 55, F(30) is 832,040, F(50) is 12,586,269,025 and F(100) is 354,224,848,179,261,915,075.
- F(1000) has 209 digits; the page prints all of them.
- Start index 1: with seeds 1 and 1 the first label is F(1) = 1, so the 10th term, F(10), is 55 just as in the classic sequence.
Index of the first seed
Textbooks disagree about whether the sequence begins at F(0) = 0 or at F(1) = 1. This tool lets you decide: the index you choose is the label of the first seed, and every later term is numbered after it. Keep seeds 0 and 1 with Start at F(0) to match the standard definition, where F(1) = 1 and F(2) = 1. If you ask for a term below the first label, for example F(0) when the list starts at F(1), the page explains the first valid number instead of guessing.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device. The numbers you type are not stored.
Related tools
The Golden Ratio Calculator splits a length into golden proportions, the same ratio the Fibonacci numbers approach. The Prime Number Generator lists primes, which is handy because a Fibonacci number F(n) can only be prime when n is prime (the one exception is F(4) = 3). The Random Number Generator picks numbers at random when you need unpredictable values.
Frequently asked questions
What is the Fibonacci sequence?
It is the list of numbers where each term is the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. The rule is F(n) = F(n-1) + F(n-2).
How do I find the n-th Fibonacci number?
Choose n-th term, type n and press Find term. With seeds 0 and 1 and the index starting at F(0), F(10) is 55, F(50) is 12,586,269,025 and F(100) is 354,224,848,179,261,915,075.
Does the sequence start at 0 or at 1?
Both conventions are used. Pick Start at F(0) with seeds 0 and 1 for the standard definition, where F(0) = 0 and F(1) = 1. Pick Start at F(1) if your course labels the first number F(1); the first seed then takes that label.
How do I generate the Lucas numbers?
Set the first seed to 2 and the second seed to 1. The list then reads 2, 1, 3, 4, 7, 11, 18, 29 and so on. Any other pair of whole numbers works the same way.
How large can the numbers get?
The page uses exact whole-number arithmetic, so nothing is rounded. F(1000) has 209 digits and is shown in full. Lists are limited to 1000 terms and seeds to 40 digits.
What is the connection to the golden ratio?
Divide a Fibonacci number by the one before it and the answer approaches the golden ratio, 1.618033988749895. The page shows F(n+1)/F(n) for your last term next to that value, and the Golden Ratio Calculator links from the form.
Is anything I type sent to a server?
No. Nothing is uploaded; every calculation runs in your browser and never leaves your device. The seeds, counts and term numbers you type are not stored either, so you can reload the page to clear them.