Sequence Calculator

What this calculator does

This is a general sequence explorer. Instead of one kind of pattern, it knows a whole shelf of them: arithmetic and geometric, but also Fibonacci, prime numbers, triangular numbers, square and cube numbers, powers of 2, factorials, and a star pattern. Pick a type, and it shows you the terms.

It is the tool to reach for when you want to see what a named sequence actually looks like, or to peek at a stretch of one. It runs right here in the browser.

Using the calculator

1. Choose a sequence type from the dropdown.
2. For arithmetic, also enter the first term and the common difference. For geometric, the first term and the common ratio. The other types need no extra settings.
3. Enter a starting position, which term to begin from, and press Calculate.

It shows a short run of terms from that position onward, so you can start anywhere in the sequence, not just at the beginning.

The sequences it can build

A quick tour of what each one is, with its opening terms:

• Arithmetic: a constant amount added each step, like 2, 5, 8, 11.
• Geometric: a constant amount multiplied each step, like 2, 6, 18, 54.
• Fibonacci: each term is the sum of the two before it, giving 0, 1, 1, 2, 3, 5, 8, 13. It shows up all over nature, in the spirals of shells and sunflower seeds.
• Prime numbers: the whole numbers with exactly two divisors, 1 and themselves, up to a limit: 2, 3, 5, 7, 11, 13.
• Triangular numbers: 1, 3, 6, 10, 15, the running totals of 1, 2, 3, 4, the number of dots that pack into a triangle.
• Square numbers: 1, 4, 9, 16, 25, each a whole number times itself.
• Cube numbers: 1, 8, 27, 64, 125, each a whole number cubed.
• Powers of 2: 1, 2, 4, 8, 16, 32, the doubling sequence at the heart of how computers count.
• Factorial: 1, 2, 6, 24, 120, each whole number multiplied down to 1, which grows terrifyingly fast.
• Star pattern: a growing arrangement of asterisks, more a visual shape than a number sequence.

When to use this, and when to use a focused one

This explorer is best for seeing and comparing the different kinds of sequence, or for the named ones like triangular, square and Fibonacci that do not have a tool of their own. But if you are working seriously with an arithmetic or a geometric sequence and you need the nth term or the running total, the focused tools go deeper. The arithmetic sequence calculator gives you the nth term and the sum, and the geometric sequence calculator lists the full run with exact-decimal precision. Use this one to browse, those to calculate.

A worked example

Say you choose triangular numbers and start from the first position. The tool shows the opening run: 1, 3, 6, 10, 15. Each one is the previous plus the next counting number: 1, then 1 + 2 = 3, then 3 + 3 = 6, then 6 + 4 = 10, then 10 + 5 = 15. That is why they are called triangular, since that many dots stack neatly into a triangle.

Questions people ask

What kinds of sequence can it show?

Arithmetic, geometric, Fibonacci, prime numbers, triangular numbers, square numbers, cube numbers, powers of 2, factorials, and a star pattern.

Can I start partway through a sequence?

Yes. The starting position lets you begin from any point, so you can look at a stretch further along, not just the first few terms.

What is the Fibonacci sequence?

A sequence where each number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13. It appears throughout nature and is closely tied to the golden ratio.

What are triangular numbers?

The running totals of the counting numbers: 1, 3, 6, 10, 15. Each is how many dots fit into a triangle with that many rows.

Should I use this or the arithmetic and geometric calculators?

Use this to explore and compare the types. For the nth term or the sum of a specific arithmetic or geometric sequence, the focused calculators do more.

References

A note on where this comes from. Each of these is a well-studied integer sequence with a standard definition, and the definitive catalogue of such sequences, including the Fibonacci, triangular, square, prime and factorial sequences, is the On-Line Encyclopedia of Integer Sequences (OEIS). For further reading, see Integer sequence.

1. The On-Line Encyclopedia of Integer Sequences (OEIS), the standard reference catalogue of integer sequences. https://oeis.org/