Fibonnaci Sequence Generator

What this calculator does

The Fibonacci sequence is one of the most famous patterns in all of mathematics: a run of numbers where each one is the sum of the two before it, giving 0, 1, 1, 2, 3, 5, 8, 13, 21 and on. This generates a stretch of them starting from a point you choose.

Tell it where to start and how many terms you want, and it lists them. It runs right here in the browser.

Using the calculator

1. Enter a range start number, roughly where in the sequence you want to begin.
2. Enter how many terms you want.
3. Press Calculate.

It finds where your start value falls in the sequence and lists that many Fibonacci numbers from there onward. Reset clears the boxes.

What the Fibonacci sequence is

The sequence starts with 0 and 1, and from then on every number is simply the two before it added together. So 0 and 1 make 1, then 1 and 1 make 2, then 1 and 2 make 3, then 2 and 3 make 5, and it rolls on: 8, 13, 21, 34, 55, 89, 144. Each number leans on its two predecessors, which is the whole of the idea.

It carries the name of Leonardo of Pisa, known as Fibonacci, who brought it to Europe in 1202 to describe an idealised rabbit population. But its real fame comes from how widely it turns up, both in nature and in a surprising link to the golden ratio, both of which are below.

The rule that builds it

Written compactly, the rule is:

F(n) = F(n − 1) + F(n − 2)

starting from F = 0 and 1. That is, each term is the sum of the previous two. It is a recurrence, meaning each new value is defined using earlier ones, and it is about as simple as a rule can be while still producing something endlessly interesting.

A worked example

Ask for 5 terms starting around 5. Since 5 is itself a Fibonacci number, the tool begins there and adds each pair as it goes:

5, 8, 13, 21, 34

Check the rule along the way: 5 + 8 = 13, then 8 + 13 = 21, then 13 + 21 = 34. Every term is the sum of the two before it, exactly as promised.

The golden ratio hiding inside

Here is the surprise that makes this sequence special. Take any Fibonacci number and divide it by the one before it, and watch what happens as you move along: 8 ÷ 5 = 1.6, then 13 ÷ 8 = 1.625, then 21 ÷ 13 ≈ 1.615, then 34 ÷ 21 ≈ 1.619. The answers dance above and below a single number and close in on it: the golden ratio, about 1.618, written with the Greek letter phi.

This is not a coincidence. Any sequence built by adding the two previous terms settles toward the golden ratio, no matter where it starts. That deep link is why Fibonacci numbers, the golden ratio, the golden rectangle and the golden section are really one story told from different angles.

Why it shows up in nature

Fibonacci numbers appear again and again in living things. The spirals of seeds in a sunflower head, the bumps on a pinecone, the arrangement of leaves around a stem, the petals on many flowers, all tend to come in Fibonacci counts. It looks magical, but there is a practical reason: arranging things by these proportions, tied to the golden ratio, packs them efficiently and gives each new leaf or seed the most room and light. Nature is not counting; it is just following the arrangement that works best, and the Fibonacci numbers fall out of it.

Questions people ask

What is the Fibonacci sequence?

A sequence starting 0, 1 in which each number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

What is the rule for finding the next term?

Add the two previous terms. In symbols, F(n) = F(n − 1) + F(n − 2).

How is it related to the golden ratio?

Divide any term by the one before it and the result closes in on the golden ratio, about 1.618, the further along the sequence you go.

Where does it appear in nature?

In sunflower seed spirals, pinecones, the arrangement of leaves, and the petal counts of many flowers, because these proportions pack growing things efficiently.

Who was Fibonacci?

Leonardo of Pisa, an Italian mathematician who introduced the sequence to Europe in his 1202 book Liber Abaci, though it was known earlier in Indian mathematics.

References

A note on where this comes from. The sequence in which each term is the sum of the two before it was introduced to Europe by Leonardo of Pisa (Fibonacci) in his Liber Abaci of 1202, through a problem on rabbit populations, though it had been described centuries earlier by Indian mathematicians such as Pingala and Virahanka. For further reading, see Fibonacci sequence.

1. Leonardo of Pisa (Fibonacci), Liber Abaci (1202), which introduced the sequence to Western Europe.
2. Earlier Indian mathematics (Pingala, c. 200 BCE, and Virahanka, c. 700 CE), where the sequence appears in the study of poetic metre.
3. The Fibonacci numbers, OEIS sequence A000045. https://oeis.org/A000045