How to use the sequence pattern finder
- Enter at least four terms of the sequence in order. Whole numbers, decimals and fractions such as 3/4 are accepted.
- Choose whether the first term is n = 1 or n = 0, which sets the numbering in the formula.
- Choose how many next terms to show.
- Read the pattern name, the formula and the next terms. The differences table shows how a polynomial was detected.
Formula
Arithmetic and polynomial: repeated differences become constant. Constant at level d means degree d.
Geometric: every term divided by the one before gives the same ratio r. a(n) = first term x r^(steps).
Linear recurrence: a(n) = p a(n-1) + q a(n-2), with p and q solved from the first terms.
Polynomial form: a(n) = sum of (n-th forward difference of the first term) x C(n - start, j)
All arithmetic is done with exact fractions, so no rounding error can fake or hide a pattern. The page needs d + 2 terms to confirm a polynomial of degree d.
Three sequences
The terms 2, 5, 10, 17, 26 have first differences 3, 5, 7, 9 and constant second differences of 2, so the pattern is quadratic with a(n) = n^2 + 1 and the next terms are 37, 50, 65, 82 and 101. The terms 3, 6, 12, 24, 48 have the constant ratio 2, so the next terms are 96, 192 and 384.
The terms 1, 1, 2, 3, 5, 8, 13 fit a(n) = 1 x a(n-1) + 1 x a(n-2), so the next are 21, 34 and 55. The terms 5, 12, 25, 44, 69, 100, 137 are quadratic with a(n) = 3n^2 - 2n + 4.
How the search works
The page tries a short list of common rules, from simplest to more complex, and reports the first that reproduces every term you entered: a constant, a polynomial found by finite differences (which includes arithmetic sequences), a geometric sequence and a second-order linear recurrence such as the Fibonacci rule. A rule is accepted only if it explains all the entered terms and leaves at least one term as a check.
Because a polynomial of degree d needs d + 2 terms to be confirmed, entering more terms makes the answer more reliable. Use more terms if the page reports no simple pattern.
Why a pattern is only a suggestion
Any finite list of numbers fits infinitely many rules. The terms 1, 2, 4, 8, 15 follow a cubic with next term 26, while 1, 2, 4, 8, 16 follow powers of two. The page picks the simplest rule it knows, which is a reasonable guess but not a proof. Rules like primes, digit manipulations or lookup tables are outside what it searches.
For well-known sequences see the Fibonacci calculator, the Collatz sequence calculator and the Pascal triangle generator. The prime number calculator covers sequences that have no simple formula.
Assumptions and limits
- Terms are given in order, with nothing missing, and are numbers or fractions of at most 30 digits.
- Only constant, polynomial up to degree 5, geometric and second-order linear recurrence patterns are searched.
- A rule must be consistent with every entered term, and a polynomial must be confirmed by at least two equal differences.
- Next terms are predictions under the chosen rule, not facts about the source of your numbers.
Frequently asked questions
Why does the page ask for at least four terms?
With only three terms almost any rule can be fitted, so there is nothing to confirm it. Four terms let the page test the rule on at least one number it was not built from.
Why did it report a cubic for 1, 2, 4, 8, 15?
Those five numbers satisfy a cubic polynomial exactly, and the simpler doubling rule fails at the fifth term. If you meant powers of two, enter the next term 16 and the geometric rule will be found.
What does n = 0 versus n = 1 change?
Only the numbering used in the formula. The same terms in the same order are described, but with n = 0 the first term is a(0), so formulas such as n^2 shift to (n + 1)^2 for the same list.
Can it find rules like primes or digit sums?
No. It looks for algebraic rules built from differences, ratios and recurrences. Sequences defined by properties of numbers need dedicated tools, such as the prime number calculator.