Recursive Sequence Calculator

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Enter your recurrence

Complete the fields required by your selected recurrence and click Generate Sequence. The calculation is processed by PHP when you submit.

Recurrence type

Starting index

Number of terms (1–100)

First initial term

Second initial term Required for Fibonacci-style and second-order recurrences.

Parameter Multiplier, ratio or coefficient.

Coefficient c2

Constant

Reset

How recursive sequences work

A recursive sequence is defined by a rule that uses one or more previous terms to calculate the next term. The initial values establish the starting point of the sequence.

Arithmetic sequence a(n) = a(n-1) + d Adds a fixed difference to each previous term.

Geometric sequence a(n) = r × a(n-1) Multiplies each previous term by a fixed ratio.

Linear recurrence a(n) = r × a(n-1) + d Multiplies the previous term and adds a constant.

Second-order recurrence a(n) = c1 × a(n-1) + c2 × a(n-2) + k Uses the two preceding terms and a constant.

Fibonacci-style sequence a(n) = a(n-1) + a(n-2) Starting with 0 and 1 gives 0, 1, 1, 2, 3, 5, ...

Factorial-style sequence a(n) = n × a(n-1) Starting at n = 1 with a(1) = 1 gives 1, 2, 6, 24, 120, ...

Frequently asked questions

Why do I need initial values?

The recurrence rule needs starting values before it can generate later terms. A first-order recurrence needs one initial value; a second-order recurrence needs two.

How many terms can I calculate?

This calculator generates between 1 and 100 terms per calculation.

Does this calculator solve every recurrence symbolically?

No. It evaluates the supported recurrence types numerically; it does not derive a closed-form solution for an arbitrary recurrence.