Enter recurrence details
Enter coefficients and initial terms as comma-separated values. Example: coefficients 1, 1 and initial terms 0, 1.
Formula used
This calculator solves a linear recurrence of order k. The coefficients c1 to ck weight earlier terms. The value d is an optional constant shift.
The starting values provide the first k known terms. After that, each new term is built from the previous terms and the constant term.
Characteristic polynomial for the homogeneous part:
How to use this calculator
- Select the recurrence order.
- Choose whether your sequence starts at index 0 or 1.
- Enter exactly that many coefficients.
- Enter the same number of starting terms.
- Add a constant term if your recurrence includes one.
- Enter the target index n to find the requested term.
- Set how many terms you want displayed in the table and graph.
- Click calculate, then export the results as CSV or PDF if needed.
Example data table
Example recurrence: an = 2an-1 - an-2 + 3, with starting terms a0 = 1 and a1 = 4.
| n | Term | Working Note |
|---|---|---|
| 0 | 1 | Starting value |
| 1 | 4 | Starting value |
| 2 | 10 | 2×4 - 1 + 3 |
| 3 | 19 | 2×10 - 4 + 3 |
| 4 | 31 | 2×19 - 10 + 3 |
| 5 | 46 | 2×31 - 19 + 3 |
Frequently asked questions
1. What is a general linear recurrence?
It is a sequence rule where each new term depends linearly on earlier terms. Coefficients scale earlier values. A constant term may also be included.
2. Why must coefficients and initial terms match the order?
An order-k recurrence needs exactly k coefficients and k starting values. Without them, the calculator cannot build the sequence correctly.
3. Can I use negative or decimal coefficients?
Yes. The calculator accepts positive, negative, and decimal inputs. This helps model many practical recurrence problems and custom sequences.
4. What does the constant term do?
The constant term adds the same amount during each recurrence step. It shifts the sequence away from a purely homogeneous pattern.
5. What is the target n value?
Target n is the sequence index you want solved. The calculator computes all needed earlier terms automatically.
6. Why is the graph useful?
The graph helps you spot growth, oscillation, decline, or stabilization. It makes recurrence behavior easier to interpret quickly.
7. What does the characteristic polynomial show?
It summarizes the homogeneous part of the recurrence. Its roots often help explain long-run sequence behavior and closed-form solutions.
8. Can I export results for reports or worksheets?
Yes. You can download a CSV table and a PDF summary. This is useful for homework, documentation, and review.