Recurrence Relation Calculator
Solve linear recurrence relations a_n = p*a_{n-1} + q*a_{n-2} with given initial conditions.
Inputs
Results
a(n) Value
55
How to Use This Calculator
- Enter the initial values a0 and a1 for the two-term recurrence.
- Set Coefficient p (for a_{n-1}) and Coefficient q (for a_{n-2}).
- Enter Target n to compute the nth term of the sequence.
- Review a(n) Value from the iterative calculation and the Closed-Form Estimate.
- Use the Closed-Form Estimate to identify the dominant growth rate (exponential, polynomial, etc.).
How the result changes with Target n
| Target n | a(n) Value |
|---|---|
| 10 | 55 |
| 35 | 9,227,465 |
| 65 | 17,167,680,177,565 |
| 90 | 2,880,067,194,370,816,000 |
What each input means
- a₀ (initial value)
- First initial condition a₀
- a₁ (initial value)
- Second initial condition a₁
- Coefficient p
- Coefficient of a_{n-1} in the recurrence a_n = p*a_{n-1} + q*a_{n-2}
- Coefficient q
- Coefficient of a_{n-2} in the recurrence a_n = p*a_{n-1} + q*a_{n-2}
- Target n
- Compute the sequence up to this index
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersa₀ (initial value) = 0, a₁ (initial value) = 1, Coefficient p = 1, Coefficient q = 1 = 5 input(s) provided
- Calculate a(n) Valuea(n) Value55 = 55
- Calculate Closed-Form EstimateClosed-Form Estimate55 = 55
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators.
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