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Category Math & Algebra

Enter your number sequence Separate numbers with commas or spaces. Supports negatives and decimals.

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About

Identifying the generating rule behind a finite sequence of numbers is a core problem in discrete mathematics and psychometric testing. Errors in pattern recognition propagate: a misidentified common difference d yields exponentially diverging predictions after just 3 - 4 extrapolated terms. This tool applies a hierarchy of detection algorithms - finite difference analysis up to order 6, ratio analysis for geometric and exponential forms, linear recurrence scanning for Fibonacci-type rules, and Newton's forward difference formula for polynomial reconstruction. It returns the simplest rule that fits your data, following Occam's razor.

The tool assumes noise-free input. If your data contains measurement error, the detected pattern may be a polynomial artifact rather than the true generating function. For sequences shorter than 3 terms, multiple valid continuations always exist. The tool selects the lowest-order polynomial consistent with the data. Provide at least 4 - 5 terms for reliable detection of quadratic or higher-order patterns.

Formulas

The primary method is finite difference analysis. Given a sequence a₁, a₂, …, a_n, the first-order differences are:

Δa_i = a_i+1 − a_i

If Δa_i is constant, the sequence is arithmetic. Otherwise, compute higher-order differences Δ^k recursively. If Δ^k becomes constant at order k, the sequence is generated by a polynomial of degree k. Newton's forward difference formula reconstructs the next term:

a_n+1 = k∑j=0 Δ^ja₀j! ⋅ ∏m (n − m)

For geometric sequences, the ratio test is applied:

r = a_n+1a_n

If r is constant for all consecutive pairs, the next term is a_n ⋅ r. For Fibonacci-type detection, we verify the recurrence a_n = a_n−1 + a_n−2 across all valid indices.

Where: a_i = the i-th term of the sequence, Δ^k = the k-th order finite difference, r = common ratio, d = common difference, n = term index.

Reference Data

Sequence Type	General Form	Example	Min Terms Needed	Detection Method
Constant	a_n = c	5, 5, 5, 5	2	0th-order difference
Arithmetic	a_n = a₁ + (n − 1)d	2, 5, 8, 11	3	1st-order constant difference
Geometric	a_n = a₁ ⋅ rⁿ⁻¹	3, 6, 12, 24	3	Constant ratio
Quadratic	a_n = an² + bn + c	1, 4, 9, 16	4	2nd-order constant difference
Cubic	a_n = an³ + …	1, 8, 27, 64	5	3rd-order constant difference
Fibonacci-type	a_n = a_n−1 + a_n−2	1, 1, 2, 3, 5	5	Recurrence relation scan
Power of n	a_n = n^k	1, 32, 243, 1024	4	Logarithmic ratio analysis
Exponential	a_n = kⁿ	2, 4, 8, 16	3	Constant ratio (special geometric)
Triangular Numbers	T_n = n(n + 1)2	1, 3, 6, 10, 15	4	2nd-order difference = 1
Alternating	a_n = (−1)ⁿ ⋅ f(n)	1, −2, 3, −4	4	Sign factor extraction
Factorial	a_n = n!	1, 2, 6, 24, 120	5	Ratio yields 2, 3, 4, 5…
Primes	p_n	2, 3, 5, 7, 11	5	Primality test on terms

Frequently Asked Questions

It depends on the pattern complexity. Arithmetic and geometric sequences need a minimum of 3 terms. Quadratic patterns require 4 terms (so that the 2nd-order difference can be confirmed constant across at least 2 values). Fibonacci-type and cubic patterns need 5 terms. As a rule of thumb, provide at least k + 2 terms where k is the suspected polynomial degree. More terms always improve confidence.

The tool applies Occam's razor: it selects the simplest pattern that fits all given terms exactly. The priority order is constant, arithmetic, geometric, Fibonacci-type, quadratic, cubic, and then higher-order polynomial. For example, the sequence 1, 2, 4 could be geometric (ratio 2) or quadratic. The tool selects geometric because it is a simpler model. If you know the intended pattern is different, add more terms to disambiguate.

Yes. The difference engine and ratio engine operate on floating-point arithmetic. Negative terms are fully supported. For geometric detection involving negative terms, the tool checks for alternating-sign geometric progressions where r < 0. Decimal values are compared with a tolerance of 1e-9 to account for floating-point rounding. Extremely large numbers (beyond 10¹⁵) may lose precision due to IEEE 754 limitations.

The tool tests a finite set of common patterns: polynomial up to degree 6, geometric, Fibonacci-type recurrences, factorial ratios, and prime sequences. If none of these models fit your data exactly, it reports no pattern. This typically occurs with sequences generated by transcendental functions (e.g., sin(n)), chaotic maps, or sequences defined by external data (e.g., digits of π). Adding more terms may help if the pattern is polynomial of degree > 6.

Finite differences will always eventually become "constant" for a finite dataset because you run out of terms. The tool guards against this by requiring the constant difference level to have at least 2 identical values, and it cross-validates the polynomial extrapolation against the original terms. For truly non-polynomial sequences like geometric or exponential, the difference table will not converge meaningfully, and the tool falls back to ratio analysis and recurrence detection.

For factorials, the tool computes consecutive ratios a_n+1 ÷ a_n and checks if they form the natural number sequence 2, 3, 4, 5… For primes, it tests whether every term in the input is prime using trial division up to √n and whether the terms match the ordered prime sequence starting from some index. Both checks require at least 4 - 5 terms to be reliable.