Exact Pell Sequence Calculator

Use P(0) = 0 and P(1) = 1, then calculate P(n) = 2P(n-1) + P(n-2) for each later index. The zero-based seeds distinguish this sequence from similarly named recurrences. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

The recurrence defines Pell numbers; the output is not a Fibonacci or Lucas term at the same index. Generate every preceding integer term and verify the final two-term update. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

The index is capped at 40 before Pell growth approaches unsafe integer precision. The selected cap leaves a clear safe-integer margin. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.