CALCULATE WITH CONFIDENCE

Sequences & Series

Recurrence Relation Solver

Generate a second-order homogeneous linear recurrence and inspect its characteristic roots.

Sequences & Series

Enter values

Private calculation in your browser
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Enter finite decimal or scientific-notation numbers, and comma-separated terms where requested. Inputs stay on this device.

Result

—

Enter valid values to see the result.

Your entries are calculated in this browser and are not submitted to 365CALCS.COM.

Feedback

Understand the subject

What is Recurrence Relation Solver?

A sequence follows an ordered rule; a series adds selected sequence terms into one total.

Use it to generate, predict, add, or test an ordered pattern while keeping its rule explicit.

The relationship

Write the rule before calculating

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

With p=1, q=1, a₀=0, a₁=1, index 10 gives 55.

Interpret with care

Important boundary

A few initial terms can fit many possible rules; a generated pattern is not proof of a unique universal formula.

Use the Mathematics collection to move between connected concepts without duplicating the calculation.

Extended domain

Characteristic roots, including complex pairs

A second-order recurrence can oscillate when its characteristic equation has complex-conjugate roots.

Try an example
CHARACTERISTIC ROOTSr1 = 1.61803398875r2 = -0.61803398875

Principal branch

ReImr1r2

Quick guide

How to use this calculator

  1. Choose the supported sequence, series, or analysis mode when the page offers one.
  2. Enter the labelled parameters or comma-separated finite terms.
  3. Read the primary result, then use the supporting classification, formula, or generated values.

Calculation method

Apply the stated sequence or series rule

aₙ=p·aₙ₋₁+q·aₙ₋₂

The calculator validates the selected family and its mathematical domain before returning a result.

Worked example

Worked example

With p=1, q=1, a₀=0, a₁=1, index 10 gives 55.

aₙ=p·aₙ₋₁+q·aₙ₋₂

Supported inputs

Precision and limits

Supported scope

General-looking pages use explicitly listed families instead of pretending to parse or prove arbitrary symbolic sequences and series.

Numerical precision

Fibonacci, Lucas, Tribonacci, Pascal, and triangular-number outputs use exact integers. Analytical and decimal calculations use finite double-precision arithmetic and normally display 12 significant digits.

Visible computational limits

Named-sequence indices are limited to 0–5000; Pascal rows to 0–200; harmonic indices to 1–1,000,000; recurrence and arithmetic-geometric generation to 1000; Taylor degree to 100; Lagrange degree to 170; sigma bounds to −100,000…100,000 with at most 200,000 steps; pattern and difference lists to 100 and 50 terms. Field labels show narrower limits where applicable.

Convergence

A convergence test applies only when its hypotheses match the entered series. Inconclusive tests and power-series endpoints are reported honestly rather than guessed.

Calculator-specific rule

This solver supports constant-coefficient, second-order homogeneous recurrences.