Analyze recurrences using coefficients, forcing terms, and initials. View roots, generated terms, and dynamic plots. Download outputs for lessons, verification, assignments, and revision work.
This calculator uses a constant-coefficient non-homogeneous linear recurrence of order k:
a(n) = c₁a(n-1) + c₂a(n-2) + ... + cₖa(n-k) + g(n)
The coefficients define the homogeneous structure. The forcing term g(n) adds the non-homogeneous contribution. This page supports constant, polynomial, exponential, trigonometric, and custom forcing sequences.
For each index n ≥ k, the calculator multiplies earlier terms by their matching coefficients, adds the forcing value, and generates the next sequence term. The characteristic polynomial of the associated homogeneous part is:
r^k - c₁r^(k-1) - c₂r^(k-2) - ... - cₖ = 0
The displayed roots come from that polynomial. Their magnitudes help interpret whether the homogeneous part tends to decay, stay balanced, or grow.
Choose the recurrence order first. The order determines how many previous terms the sequence uses.
Enter the recurrence coefficients in descending lag order. For a second-order rule, type values for a(n-1) and a(n-2).
Enter the initial sequence values from a(0) through a(k-1). These seed the recurrence.
Select a forcing type and supply its parameters. Use custom mode when you already know forcing values by index.
Choose how many terms to generate, then press the solve button. The result appears above the form, followed by a graph and downloadable output table.
Non-homogeneous linear recurrences appear in algorithm analysis, digital signal models, discrete control, financial schedules, and many counting problems. A useful solver should do more than return one number. It should reveal structure, growth behavior, and the effect of external input terms.
This calculator generates a full sequence table, not just the last term. That helps you inspect whether the forcing term dominates, whether oscillation appears, and whether early values strongly influence later behavior. The chart makes trend changes easier to notice during homework checks and revision.
The characteristic polynomial belongs to the associated homogeneous recurrence. Its roots summarize the internal behavior of the recurrence before the forcing term is added. When a dominant root has magnitude below one, the homogeneous contribution usually fades. When it exceeds one, growth can become pronounced.
Because many textbook questions use constant, polynomial, exponential, or sinusoidal forcing, this page includes all of those options. Custom forcing is also available when your problem supplies values directly by index. The export tools are useful for reports, worksheets, and comparing multiple recurrence experiments.
| Item | Example value |
|---|---|
| Order | 2 |
| Coefficients | 2, -1 |
| Initial values | 1, 3 |
| Forcing type | Constant |
| Constant forcing | 1 |
| Generated terms | 12 |
| Recurrence | a(n) = 2a(n-1) - a(n-2) + 1 |
It means the recurrence includes an added forcing term, written as g(n). Without that extra part, the relation would be homogeneous.
They multiply earlier sequence terms. Together, they define the core linear rule that produces each new value from previous values.
The recurrence cannot start itself. Initial values provide the first known terms, which the solver uses to build every later term.
They describe the associated homogeneous behavior. Their sizes and possible complex parts help explain decay, growth, repeated patterns, or oscillation.
Yes. The calculator accepts decimal entries for coefficients, initial values, and forcing parameters, which is useful for applied modelling problems.
You enter forcing values by index. If the sequence runs beyond your list, the last forcing value is reused for remaining terms.
It focuses on numerical sequence generation and homogeneous-root analysis. That makes it practical for checking many recurrence problems quickly.
Exports help with assignments, class notes, verification steps, and record keeping when you compare several recurrence setups.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.