Number Sequence Generator

What a number sequence generator does

A number sequence is an ordered list of numbers built by a rule. This tool does three related jobs: it generates the terms of well-known sequences, it tries to identify an unknown sequence you paste in, and it builds custom arithmetic or geometric progressions from your own starting values. Each built-in sequence comes with its defining formula and, where one exists, its reference number in the On-Line Encyclopedia of Integer Sequences (OEIS), so you can verify the math against an authoritative catalogue rather than taking the output on faith.

The sequences it knows, and their rules

The generator ships with eleven named patterns. The recursive and closed-form rules behind them are:

The arithmetic option uses a(n) = a₀ + n·d and the geometric option uses a(n) = a₀ · rⁿ, where you choose the start, the common difference d, or the common ratio r.

A worked example

Pick Triangular Numbers and the tool generates the first 20 terms. Term 5 is T(5) = 5 × 6 / 2 = 15; term 6 is 6 × 7 / 2 = 21. Notice each term is the previous one plus its own index: 15 + 6 = 21. That is the visual meaning of "triangular" — stacking rows of 1, 2, 3, 4… dots. Switch to the geometric tab with start 1 and ratio 2 and you get the powers of two; change the ratio to 3 and the same machinery produces 1, 3, 9, 27, 81. One formula, many sequences.

Identifying a mystery sequence

Paste at least three comma-separated numbers into the Identify tab and the tool checks them against every built-in pattern, reporting each match with its formula and OEIS code. Give it 2, 1, 3, 4, 7 and it recognises the Lucas numbers; give it 1, 3, 6, 10 and it returns Triangular. For arithmetic and geometric guesses it infers the step or ratio from your first two numbers, so 5, 8, 11, 14 is identified as an arithmetic progression with difference 3. If nothing matches within a small tolerance, it says so honestly rather than inventing a rule — a useful signal that your sequence may be irregular, mistyped, or simply outside the built-in set.

Use cases

• Homework and self-study. Check whether your hand-computed terms are right, and see the closed-form formula next to the recurrence.
• Test data and puzzles. Generate a clean list of squares, primes, or factorials to seed unit tests, spreadsheets, or programming exercises, then copy it with one click.
• Spotting a pattern. When you have a few numbers from a problem and suspect there's a rule, the Identify tab is faster than guessing.
• Teaching. Demonstrate how Fibonacci and Lucas share a recurrence but differ only in their seeds, or how triangular numbers are partial sums of the integers.

Tips and common mistakes

• Three numbers is the minimum to identify. Two numbers fit infinitely many rules; the tool requires at least three and ignores anything that isn't a valid number.
• Factorials and powers grow explosively. By the twentieth term a factorial is astronomically large, and beyond a certain point standard floating-point arithmetic loses exactness. Treat very large terms as approximate.
• The Identify tab only knows its built-in list. "No match" means none of these eleven patterns fit — not that your sequence is meaningless. For an exhaustive search, look the terms up directly in the OEIS.
• Geometric ratios can be fractional. A ratio of 0.5 produces a halving sequence; the field accepts decimals, so you are not limited to whole-number growth.

Everything is computed in your browser as you type — no input is sent anywhere — so you can generate and identify sequences offline.