Editing Pell number (section)

{{Short description|Number used to approximate the square root of 2}}
{{distinguish|Bell number}}

[[File:Silver spiral approximation.svg|thumb|The sides of the [[square]]s used to construct a silver spiral are the Pell numbers]]

In [[mathematics]], the '''Pell numbers''' are an infinite [[integer sequence|sequence of integers]], known since ancient times, that comprise the [[denominator]]s of the [[closest rational approximation]]s to the [[square root of 2]]. This [[sequence]] of approximations begins {{sfrac|1|1}}, {{sfrac|3|2}}, {{sfrac|7|5}}, {{sfrac|17|12}}, and {{sfrac|41|29}}, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the '''companion Pell numbers''' or '''Pell–Lucas numbers'''; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

Both the Pell numbers and the companion Pell numbers may be calculated by means of a [[recurrence relation]] similar to that for the [[Fibonacci number]]s, and both sequences of numbers [[exponential growth|grow exponentially]], proportionally to powers of the [[silver ratio]] 1&nbsp;+&nbsp;{{sqrt|2}}. As well as being used to approximate the square root of two, Pell numbers can be used to find [[square triangular number]]s, to construct [[integer]] approximations to the [[right isosceles triangle]], and to solve certain [[combinatorial enumeration]] problems.<ref>For instance, Sellers (2002) proves that the number of [[perfect matching]]s in the [[Cartesian product of graphs|Cartesian product]] of a [[path graph]] and the [[graph (discrete mathematics)|graph]] ''K''<sub>4</sub>&nbsp;−&nbsp;''e'' can be calculated as the product of a Pell number with the corresponding Fibonacci number.</ref>

As with [[Pell's equation]], the name of the Pell numbers stems from [[Leonhard Euler|Leonhard Euler's]] mistaken attribution of the equation and the numbers derived from it to [[John Pell (mathematician)|John Pell]]. The Pell–Lucas numbers are also named after [[Édouard Lucas]], who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are [[Lucas sequence]]s.