Recurrence Relation Calculator

Inputs

Define order, coefficients, constant term, and initial terms.

Numeric iteration up to n = 2000.

Order (k)

k = number of previous terms used.

Target n for T(n)

Example: 10 returns T(10).

Compute table up to n

Must be ≥ max(k−1, target n).

Constant term (c)

Use 0 for homogeneous recurrence.

Rounding decimals

Affects table, exports, and display.

Coefficients

For T(n) = a1·T(n−1) + a2·T(n−2) + … + ak·T(n−k) + c

a1

Multiplier for T(n−1).

a2

Multiplier for T(n−2).

Initial terms

Provide T(0) … T(k−1).

T(0)

T(1)

Tip: Set k=2, a1=1, a2=1, c=0, T(0)=0, T(1)=1 for Fibonacci.

Example data table

Example: Fibonacci recurrence (k=2, a1=1, a2=1, c=0, T0=0, T1=1).

n	T(n)
0	0
1	1
2	1
3	2
4	3
5	5
6	8
7	13

Formula used

This calculator evaluates a linear recurrence of order k: T(n) = a1·T(n−1) + a2·T(n−2) + … + ak·T(n−k) + c.

It starts from the given initial terms T(0)…T(k−1) and iterates forward to compute any requested T(n). For k=1 and k=2 homogeneous cases, it also shows standard closed-form notes.

How to use this calculator

Choose the order k of your recurrence.
Enter coefficients a1…ak and the constant term c.
Provide initial terms T(0)…T(k−1).
Set the target n and the maximum table range.
Press Submit to view T(n), table, and export buttons.

FAQs

1) What types of recurrences does this support?

It supports linear recurrences with constant coefficients, up to order six, plus an optional constant term. You can model Fibonacci-like sequences, growth/decay sequences, and many classroom examples.

2) What does the order k mean?

Order k means each new term uses the previous k terms. For k=1 you use one previous term; for k=2 you use two, and so on. You must enter k initial values.

3) What is the role of the constant term c?

The constant term shifts each step by a fixed amount. When c=0, the recurrence is homogeneous. When c≠0, it is non-homogeneous, which commonly appears in “plus a fixed cost” processes.

4) Does it give an exact closed form?

It always computes numeric values by iteration. For first-order and second-order homogeneous cases, it also shows standard closed-form patterns and approximate constants to help you verify work.

5) Why is there a limit of n=2000?

Very large n can be slow in a browser-hosted environment. A cap keeps the calculator responsive. If you need larger ranges, export CSV and compute further values in a math tool.

6) Can I use negative or decimal coefficients?

Yes. Coefficients and initial terms accept decimals and negatives. This is useful for alternating sequences, damped systems, or modeling with weighted averages. Rounding control helps format outputs.

7) How do CSV and PDF exports work?

After a successful calculation, the page stores the latest table in your session. The download buttons then generate a CSV file or a simple one-page PDF summarizing the recurrence and values.

8) What if results become extremely large?

Large coefficients or fast-growing recurrences may exceed floating precision and show Infinity or NaN. Reduce n, scale inputs, or use big-number libraries for exact integer arithmetic when required.