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Recurrence relation solver

Generate the sequence of any linear recurrence a(n) = c₁·a(n−1) + … + cₖ·a(n−k). Enter the coefficients and the matching initial values, and it unrolls the sequence term by term — Fibonacci, Lucas, Pell and any custom order-k relation you define, with decimals and negatives supported.

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Recurrence relation solver is free to use as often as you like, directly from this page. You will find it under Sequences & series, with Absolute Value Equation Solver and Absolute Value Inequality Solver for the neighbouring cases.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Recurrence relation solver do?

Generate the sequence of any linear recurrence a(n) = c₁·a(n−1) + … + cₖ·a(n−k). Enter the coefficients and the matching initial values, and it unrolls the sequence term by term — Fibonacci, Lucas, Pell and any custom order-k relation you define, with decimals and negatives supported.

What does a concrete case look like?

aₙ = aₙ₋₁ + aₙ₋₂, a₀ = 0, a₁ = 1 → Fibonacci, a₁₀ = 55 — the tool shows every step in between, not just the final figure.

When would I actually use this?

Finding the nth term without listing every one before it, summing a long run in one step, and recognising which family a sequence belongs to.

What is the most common mistake?

Starting the index at the wrong end. Whether the first term is a₀ or a₁ shifts every result by one position, and the two conventions are both common.

What else is filed next to it?

Absolute Value Equation Solver, Absolute Value Inequality Solver and Bernoulli ODE Solver share its section. They are not variants of it — being filed together is not the same as being alike — but that is where to look if this turned out not to be the tool you wanted.

Where do the figures come from?

The closed forms are the standard ones, and the tool states the indexing convention it uses so a result can be compared against a textbook without ambiguity.

Further reading

All guides →
ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.