Number Sequence Calculator

Enter a sequence of numbers to detect the pattern, find the formula, and predict the next terms.

Input & Options

Enter sequence (comma or space separated)

Terms to Predict

Index Start (n)

Starts at 1 (a₁, a₂, ...) Starts at 0 (a₀, a₁, ...)

Treat input as integers only

Enable advanced detection

Common Number Sequences

Number sequences are lists of numbers arranged in a specific order, following a certain rule or pattern. This calculator excels at identifying several common types:

• Arithmetic Sequence: A sequence where the difference between consecutive terms is constant. This constant is called the common difference (d). Example: 5, 9, 13, 17... (d=4).
• Geometric Sequence: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). Example: 3, 9, 27, 81... (r=3).
• Fibonacci Sequence: A famous sequence where each number is the sum of the two preceding ones, usually starting with 0 and 1. Example: 0, 1, 1, 2, 3, 5, 8...
• Prime Sequence: A sequence consisting of consecutive prime numbers, which are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Example: 2, 3, 5, 7, 11...
• Polynomial Sequence: A sequence whose nth term is given by a polynomial in n. Quadratic sequences (like squares: 1, 4, 9, 16...) and cubic sequences are common examples.

How This Sequence Detector Works

Our tool employs a multi-stage analysis pipeline to determine the most likely pattern behind your sequence:

1. Difference and Ratio Analysis: It first calculates the differences and ratios between consecutive terms. If the first-order differences are constant, it's an arithmetic sequence. If the ratios are constant, it's geometric. It continues to check second and third-order differences to detect polynomial patterns.
2. Pattern Matching: The sequence is checked against well-known sequences like Fibonacci numbers, primes, factorials, and triangular numbers.
3. Polynomial Fitting: If simpler methods fail, the calculator attempts to fit a polynomial of the lowest possible degree to the data points. It derives a formula like a_n = An² + Bn + C.
4. Recurrence Relation Detection: For more complex patterns, the advanced detector tries to find a linear recurrence relation, where each term is a linear combination of several preceding terms (e.g., a_n = 2a_n-1 - a_n-3).

When Sequence Detection Fails — Why and What to Try Next

Sometimes, a pattern may be too complex, too short, or ambiguous. Here are common reasons for failure:

• Insufficient Terms: With only two or three numbers, multiple patterns can fit perfectly. For reliable detection, provide at least 4-5 terms.
• Complex or Obscure Patterns: The sequence might follow a rule not included in our standard detection algorithms (e.g., related to digits, specific number theory concepts, or external data).
• Typographical Errors: A single wrong number can throw off the entire analysis. Double-check your input for accuracy.

If no pattern is found, try providing more terms. You can also try searching for the sequence in the On-Line Encyclopedia of Integer Sequences (OEIS), a massive database of number sequences.

Frequently Asked Questions

Please see the structured data in the page header for a list of frequently asked questions and their answers, optimized for search engines.

This tool provides algorithmic pattern detection and is best used for puzzles, study, and exploration. It may not discover the author’s intended pattern in ambiguous cases.