Pell Number Calculator

Enter the Nth term to calculate the corresponding Pell number. The Pell number sequence is defined by the recurrence relation P_n = 2P_n-1 + P_n-2, with P₀ = 0 and P₁ = 1.

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Pell Number Formula

The Pell sequence is a recursive integer sequence in which each term is built from the two terms before it. This calculator is useful when you want the value at a specific term, check whether a number appears in the sequence, or verify manual recurrence calculations.

P_n = 2P_{n-1} + P_{n-2}

The sequence starts from these base values:

P_0 = 0,\quad P_1 = 1

From there, each new term is found by doubling the previous Pell number and adding the one before that. Because the definition is recursive, the sequence grows quickly as the term index increases.

How to Use the Pell Number Calculator

• Enter a term index to find the Pell number at that position.
• Use the result to confirm hand calculations from the recurrence relation.
• If reverse lookup is available, enter a Pell value to identify its matching term in the sequence.
• Be careful with indexing: this calculator uses the standard convention that starts at term 0.

First Pell Numbers

This table is especially helpful for spotting the indexing pattern. Some references begin counting from 1 instead of 0, which shifts every label by one term.

Closed-Form Expression

Although the recurrence is the most natural way to define the sequence, Pell numbers also have a direct formula. This is useful for theoretical work and for understanding the growth rate of the sequence.

P_n = \frac{(1+\sqrt{2})^n - (1-\sqrt{2})^n}{2\sqrt{2}}

This expression shows why Pell numbers increase rapidly: the dominant growth comes from the factor involving the silver ratio.

\frac{P_{n+1}}{P_n} \to 1+\sqrt{2}

Example Calculation

To find the 10th Pell number, start from the base values and apply the recurrence repeatedly until you reach the target term. The values near the end of the chain are 408, 985, and 2378, so the 10th Pell number is:

P_{10} = 2(985) + 408 = 2378

This is the same process the calculator automates, which makes it much faster and reduces arithmetic mistakes for larger terms.

Useful Properties

• Every Pell number is an integer even though the closed-form expression contains radicals.
• The sequence alternates in parity: even, odd, even, odd, and so on.
• Each term is larger than the one before it after the first two terms, so the sequence is strictly increasing for positive indices.
• The values become large very quickly, so big term numbers may require arbitrary-precision arithmetic in software implementations.

A helpful identity for checking neighboring terms is:

P_{n+1}P_{n-1} - P_n^2 = (-1)^n

Identities like this are useful when validating recurrence logic, building algorithms, or checking whether a generated sequence is correct.

Why Pell Numbers Matter

Pell numbers appear in number theory, recurrence relations, continued fractions, and approximation problems involving square roots. They are closely connected to expressions involving the square root of 2 and to integer relationships of the form:

x^2 - 2y^2 = \pm 1

Because of these connections, Pell numbers show up in algebra courses, discrete mathematics, computational math, and sequence-analysis problems.

Practical Notes for Calculator Users

• Use small terms first if you are learning the pattern of the sequence.
• For larger terms, rely on the calculator rather than manual recursion to avoid compounding errors.
• If your expected answer does not match, first check whether you started at term 0 or term 1.
• When comparing answers across textbooks or websites, confirm the indexing convention before assuming one result is wrong.