Compute recurrence terms, characteristic roots, and closed forms. Handle repeated roots and polynomial forcing cases. Download tables, review steps, and test sequence assumptions confidently.
| Setting | Example Value |
|---|---|
| Coefficients | 1, 1 |
| Initial terms | 0, 1 |
| Start index | 0 |
| Target n | 10 |
| Generated terms | 15 |
| Polynomial forcing | 0, 0, 0, 0 |
| Exponential forcing | A = 0, r = 2 |
| Interpretation | Classic Fibonacci-style recurrence |
This calculator evaluates a linear recurrence of order k.
General recurrence: a(n) = c1·a(n-1) + c2·a(n-2) + ... + ck·a(n-k) + f(n)
Forcing function: f(n) = p0 + p1·n + p2·n² + p3·n³ + A·r^n
Characteristic equation: r^k - c1·r^(k-1) - c2·r^(k-2) - ... - ck = 0
When forcing is zero, the homogeneous solution depends on the characteristic roots. Repeated roots create extra n factors. Complex roots create oscillation. The calculator estimates roots numerically and uses them for growth analysis.
Linear recurrences appear in counting, algorithm analysis, and discrete modelling. They describe how each term depends on earlier terms. This structure makes them useful in pure mathematics and applied work. A fast calculator helps you test patterns before writing a proof or a program.
This tool handles fixed-order recurrences with custom initial values. It also supports a forcing function. You can include constants, polynomial terms, or an exponential driver. That makes it useful for homogeneous and non-homogeneous sequence studies.
The characteristic equation is central. Its roots explain the long-run shape of the sequence. Real roots often drive steady growth or decay. Complex roots can create oscillation. Repeated roots add extra powers of n to the solution form.
Start with the target term. Then inspect the generated table. Check the forcing column to see how outside input changes later values. Next, review the root table. The largest root modulus usually gives the main growth signal for the homogeneous part.
Use forcing when the sequence is driven by an outside pattern. A constant can model a steady added amount. A polynomial can model systematic trend. An exponential driver can model scaled growth. These options are helpful in classroom exercises and recurrence experiments.
A recurrence calculator saves time and reduces algebra mistakes. It lets you compare coefficient choices quickly. It also helps validate hand work. The export tools make it easy to share tables, attach them to notes, or keep a record of tested cases.
Many textbook problems use second-order rules. Fibonacci-style growth is one example. Divide-and-conquer algorithms often lead to structured recurrences too. Financial saving plans, population models, and queue estimates can also produce linear relations. Testing several starting values helps reveal how sensitive the sequence is.
Exported tables help with homework checks, reports, and revision sheets. A saved CSV supports spreadsheet work. A saved PDF is useful for clean printing. These small tools make the calculator easier to use in practical study workflows.
A linear recurrence defines each term from earlier terms using fixed coefficients. Many famous sequences, including Fibonacci-type sequences, follow this pattern.
An order k recurrence needs k starting values. Those values anchor the sequence and let every later term be computed.
The forcing function adds an outside input at each index. It changes the recurrence from homogeneous to non-homogeneous and can model drift, trend, or driven growth.
The characteristic equation helps identify root structure. Those roots explain growth, decay, oscillation, and repeated-root behavior in the homogeneous solution.
Yes. It estimates repeated roots numerically and shows their multiplicities. Repeated roots imply extra powers of n in the solution basis.
The roots are found numerically, so displayed values are approximations. Exact symbolic closed forms are harder, especially with repeated or complex roots.
It gives a quick growth signal for the homogeneous part. Values above one suggest growth. Values below one suggest decay.
Yes. Use the built-in CSV and PDF buttons after calculation. They export the summary and generated sequence table for review or sharing.
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.