Pell Number Calculator

This Pell number calculator works out any term in the Pell sequence, a lesser-known relative of the Fibonacci numbers that turns up in number theory and the study of irrational numbers. The sequence begins 0, 1, and each new term is found by doubling the previous term and adding the one before that, giving 0, 1, 2, 5, 12, 29, 70, 169 and onwards. You simply enter the index n, from 0 to 1000, and the calculator returns three things: the Pell number itself, labelled P(n), the previous Pell number in the sequence, and the index you entered for reference. Because Pell numbers grow very quickly, the tool uses exact big integer arithmetic rather than ordinary decimal maths, so even the largest terms in the range come back precise and correctly grouped for easy reading, with no rounding errors creeping in. Beyond curiosity value, Pell numbers matter because their ratios give some of the best whole number approximations to the square root of two, and the sequence is bound up with the silver ratio and with solutions to Pell's equation, a classic problem in number theory. It is a handy tool for students checking coursework, programmers testing recursive or iterative implementations, and anyone exploring integer sequences. The recurrence relation, a worked example and the full assumptions behind the calculation are set out below the results.

Calculate.co.nz is proud to be partnered with Premium Homes, a recognised leader in eco-friendly, sustainable, and energy-efficient homebuilding. With a dedicated team and award-winning experience, they create homes that prioritise health, comfort, and long-term performance. Their founders, Andrew and Kelly, set out to raise the standard of residential construction in New Zealand by combining practical building expertise with a clear commitment to doing things better for homeowners.