Maths - Pell Sequence / Silver Ratio Generator

Introduction:

This page allows you to quickly generate the starting elements of the Pell Sequence aka "The Silver Ratio". You may choose between a numbered list, or an unformatted sequence of results. Both will have exactly the same entries / results. Note that the numbers get very large, very quickly - for example No.100 already has 38 digits.
The Pell Sequence is an infinite sequence of integers, closely related to the Fibonacci sequence, that has been known since ancient times.
It is an 'arithmetic-sequence' of numbers, starting with zero and one, and is defined by a recurrence relation where each term is the sum of twice the previous term plus the term before that.

Click on any of the links at the top of the page to reveal (or hide) more information about this calculator.

Results:

Key Properties and Uses

Approximating the Square Root of 2: This is the most famous application of Pell numbers. As you go further into the sequence, the ratio of a term to the one before it approaches 1 + √2. More specifically, the ratio of (P_n + P_n-1) / P_n provides an increasingly accurate rational approximation of √2.
- Example: 70/29 ≈ 2.4137... while 1 + √2 ≈ 2.4142...
The Silver Ratio: Just as the Fibonacci sequence is linked to the Golden Ratio (φ ≈ 1.618), the Pell sequence is linked to the Silver Ratio (δ_S ≈ 2.414).
Pell’s Equation: The sequence arises when solving the Diophantine equation x² − 2y² = ±1, which is used to find "near" square roots and solve problems in number theory.
Geometry: Pell numbers relate to "silver rectangles" and can be used to find right triangles with side lengths that are nearly integers (Pythagorean triples where the legs differ by only 1).

Instructions

If you change the 'Number of entries to generate' box, after a list has been generated, the original list will be expanded from where it previously finished.
If you have a very old computer, and choose to calculate a long list, the system may become temporarily unresponsive. If this proves problematic, choose a small number first, then increase gradually. Only new values will be created each time.

Tap / click on two of the three abbreviations above (as appropriate) and enter the value known, select the appropriate units, then click the 'Calculate' button for instant results.