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Pell Number Calculator

Find any Pell number Pₙ.

How it is calculated

Pell numbers follow P(n) = 2 × P(n−1) + P(n−2) with P(0) = 0 and P(1) = 1. The tool builds the sequence term by term in a loop, starting from these two values, and returns the term at position n.

n must be a whole number from 0 to 40; a decimal is cut to its whole part and anything outside the range gives no result. The first term is n = 0, so the sequence reads 0, 1, 2, 5, 12, 29, 70, 169, 408 and so on. The limit of 40 keeps every value exact. The ratio of consecutive terms approaches 1 + √2 ≈ 2.414.

Worked example

Find P(8). Start with P(0) = 0 and P(1) = 1. P(2) = 2×1 + 0 = 2; P(3) = 2×2 + 1 = 5; P(4) = 2×5 + 2 = 12; P(5) = 2×12 + 5 = 29; P(6) = 2×29 + 12 = 70; P(7) = 2×70 + 29 = 169; P(8) = 2×169 + 70 = 408.

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Common mistakes

  • Counting from 1. The first term is P(0) = 0, so the eighth term in a list is P(7) = 169.
  • Mixing up Pell numbers with Fibonacci numbers. Pell uses a factor of 2 on the latest term, so it grows much faster.
  • Confusing Pell numbers with the "Pell-Lucas" (companion) numbers 2, 2, 6, 14, 34 and so on, which start differently.
  • Entering n above 40 and expecting an answer; the tool refuses so it does not give inexact values.

About this calculator

Pell numbers follow the recurrence P₀=0, P₁=1, Pₙ = 2Pₙ₋₁ + Pₙ₋₂, generating 0, 1, 2, 5, 12, 29, 70, 169 … Their defining property is that the ratio Pₙ/Pₙ₋₁ gives increasingly accurate rational approximations of √2 — the same role continued fractions play, since Pell numbers are literally the denominators (with companion Pell numbers as numerators) of √2's best rational approximants.

They're named for their connection to Pell's equation x² − 2y² = ±1, a classical Diophantine equation studied since antiquity (despite the name, John Pell had little to do with it — Euler misattributed it). Number theorists, cryptography researchers exploring Diophantine-based schemes, and recreational mathematicians use Pell numbers when studying integer solutions and square-root approximation.

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Comments (1)

  • Editorial team

    Pell numbers give a neat way to approximate the square root of 2. Add a term to the one before it and divide by the term itself: for n = 6, the tool gives 70 and the previous term is 29, so 99 ÷ 70 is about 1.41429, close to the true 1.41421. Each step gets closer. Try other n and compare. Do you meet Pell numbers in number theory, in puzzles or elsewhere?

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