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Recurrence Relation Generator

A rule that builds each term from the ones before it, and the sequence that rule produces. Fibonacci is the famous case — each term the sum of the previous two — but the same shape covers any set of coefficients, and the ratio of consecutive terms always drifts towards the largest root of the characteristic equation. Only the first few terms are ever free; everything after them is forced.

What this generator does

Draws coefficients and starting terms, then computes the sequence forward. It reports the ratio of the last two terms, which for a Fibonacci-style rule approaches the golden ratio and in general approaches the largest characteristic root.

How to use this tool

  1. Choose the order — how many previous terms each new one uses.
  2. Choose how many terms to generate.
  3. Continue the sequence yourself and compare.
  4. Watch the ratio of consecutive terms settle as the sequence grows.

Understanding the controls

Order
Between 1 and 3 — how many earlier terms each new one depends on. Order two is the Fibonacci shape.
How many terms
Between 5 and 20. Twenty terms is enough for the ratio to settle visibly.
Seed
Reproduces the same coefficients and starting terms.

Common use cases

  • Sequence continuation exercises with a stated rule
  • Showing where the golden ratio comes from
  • Test data for anything consuming a recurrence
  • Introducing characteristic roots through the ratio rather than the algebra
  • Reproducing the same sequence from a seed

How this generator works

Terms after the starting ones are computed directly from the recurrence. The check recomputes every one of them from its predecessors and requires an exact match, confirms the sequence begins with the stated starting terms, and rejects any run where a term overflowed.

Randomness and fairness

The coefficients and starting terms are random; everything after them is forced by the rule. Seeded sequences reproduce exactly and are therefore explicitly not cryptographically secure, and unseeded ones use the browser's cryptographically secure generator.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

  • Coefficients are small positive integers, so the sequences always grow and never oscillate.
  • No closed form is derived — the ratio hints at the characteristic root without solving for it.
  • Twenty terms is the ceiling, above which the values become unwieldy.
  • Non-linear recurrences and those with a non-constant term are not generated.
  • Sequences are not stored between visits.

Privacy and your data

Coefficients and terms are computed in your browser. Nothing about the sequence or your seed leaves the page.

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