Enter your sequence type (Fibonacci, Lucas, Padovan, or custom), set your initial values and recursive coefficients, then specify the term number n to compute. The Recursive Sequence Calculator returns the nth term, the full sequence list, and the convergence ratio — with a chart showing how the sequence grows term by term.
Value of Term n
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Convergence Ratio (aₙ / aₙ₋₁)
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Terms Generated
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Recurrence Formula
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A recursive sequence (or recurrence relation) defines each term based on one or more preceding terms rather than giving a direct formula for the nth term. Every recursive sequence needs at least one base case (starting value) and a recursive rule. The Fibonacci sequence — where each term equals the sum of the two before it — is the most famous example.
A recursive formula computes the nth term by working forward from initial values one step at a time, so you must compute all prior terms. An explicit (or closed-form) formula gives you the nth term directly from n without computing intermediate terms. For example, Binet's formula gives the nth Fibonacci number directly using the golden ratio. Both approaches yield the same values but differ in computational strategy.
Look for a pattern where each term depends on previous terms rather than on its position alone. Common signs include a constant ratio between consecutive terms (geometric), a constant difference (arithmetic), or more complex relationships like each term equaling the sum of the two or three before it. If you can express aₙ in terms of aₙ₋₁, aₙ₋₂, etc., it's recursive.
The Fibonacci sequence starts with 0, 1 and each subsequent term is the sum of the two before it (0, 1, 1, 2, 3, 5, 8, …). The Lucas sequence follows the same 2nd-order rule but starts with 2, 1 (2, 1, 3, 4, 7, 11, …). The Padovan sequence is a 3rd-order recurrence starting with 1, 1, 1 where aₙ = aₙ₋₂ + aₙ₋₃ (1, 1, 1, 2, 2, 3, 4, 5, 7, …). Both Fibonacci and Lucas converge toward the golden ratio φ ≈ 1.618.
The convergence ratio is the value of aₙ / aₙ₋₁ as n grows large. For Fibonacci and Lucas sequences this ratio approaches the golden ratio (φ ≈ 1.618033…). For Padovan sequences it converges to the plastic constant (≈ 1.32471…). A stable ratio indicates the sequence is converging; diverging or oscillating ratios suggest unbounded or oscillatory behavior depending on the coefficients chosen.
Recursive formulas work well for sequences where each term depends on a finite number of prior terms with a fixed rule. They handle arithmetic, geometric, Fibonacci-type, and many polynomial sequences. However, not every sequence can be expressed with a finite-order linear recurrence — some require non-linear rules, and others (like random sequences) have no useful recursive structure.
The main limitations are computational: calculating the nth term requires computing all n−1 prior terms, making it slow for very large n. Numbers can also grow extremely fast (Fibonacci at n=50 already exceeds 10¹⁰), potentially causing overflow. For large n, a closed-form solution is more efficient. This calculator supports n up to 50 to keep results accurate and manageable.
Select 'Custom 2nd-Order Recurrence' or 'Custom 3rd-Order Recurrence' from the Sequence Type menu. Enter your starting values (a₀, a₁, and a₂ for 3rd-order), then set coefficients p and q. The 2nd-order formula used is aₙ = p·aₙ₋₁ + q·aₙ₋₂, and the 3rd-order formula is aₙ = aₙ₋₁ + p·aₙ₋₂ + q·aₙ₋₃. Adjust these values and the tool will compute the full sequence up to your chosen term n.