Calculator Inputs
Enter coefficients and starting values as comma-separated lists.
Formula Used
General form:
an = c1an-1 + c2an-2 + ... + ckan-k. The characteristic equation is rk - c1rk-1 - ... - ck = 0.
How to Use This Calculator
- Enter coefficients in the exact order used by previous terms.
- Enter the same number of initial terms as coefficients.
- Choose the starting index that matches your notation.
- Set the target index for the term you want.
- Choose how many rows you want in the table and graph.
- Press the solve button to generate the term table, graph, roots, and exports.
Example Data Table
| Sequence | Coefficients | Initial terms | Starting index | Sample target | Expected value |
|---|---|---|---|---|---|
| Fibonacci | 1, 1 | 0, 1 | 0 | a10 | 55 |
| Lucas | 1, 1 | 2, 1 | 0 | a10 | 123 |
| Tribonacci | 1, 1, 1 | 0, 0, 1 | 0 | a10 | 81 |
FAQs
1) What recurrence types does this solver handle?
It solves linear recurrences with constant coefficients using comma-separated coefficients and starting values. It generates terms numerically, shows the characteristic polynomial, approximates roots, and evaluates the requested nth term.
2) Do coefficients match previous terms in order?
Yes. Enter coefficients as c1, c2, ..., ck for an = c1an-1 + c2an-2 + ... + ckan-k. Initial terms must match that order length.
3) Can I start indexing at 1 instead of 0?
Yes. Set the starting index to 1 if your given values begin with a1. The table, plotted points, and step labels follow that chosen indexing system.
4) Are decimal and negative inputs allowed?
Yes. Coefficients and initial values may be decimal or negative. That makes the tool useful for damped, oscillating, weighted, or alternating sequences found in applied mathematics.
5) Why are characteristic roots important?
Characteristic roots help reveal growth, decay, oscillation, and repeated behavior. A dominant root magnitude above 1 often suggests growth, while magnitudes below 1 usually suggest decay.
6) Can this page solve non-linear recurrences?
No. This page is designed for linear recurrences with constant coefficients. Non-linear, variable-coefficient, or piecewise recurrences usually need different numerical or symbolic methods.
7) Why can values become huge very quickly?
Many recurrences grow exponentially when the dominant root magnitude exceeds 1. Large coefficients and higher-order rules can create very large outputs after only a moderate index.
8) What do the CSV and PDF files contain?
The CSV includes the generated table. The PDF includes the recurrence, target term, characteristic equation, root summary, and the same generated table for reporting or review.