Editing Pell number (section)

== Pell–Lucas numbers ==

The '''companion Pell numbers''' or '''Pell–Lucas numbers''' are defined by the [[recurrence relation]]
:<math>Q_n=\begin{cases}2&\mbox{if }n=0;\\2&\mbox{if }n=1;\\2Q_{n-1}+Q_{n-2}&\mbox{otherwise.}\end{cases}</math>

In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell–Lucas number to the Pell–Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and {{nowrap|1=82 = 2 × 34 + 14 = 70 + 12.}} The first few terms of the sequence are {{OEIS|id=A002203}}: [[2 (number)|2]], 2, [[6 (number)|6]], [[14 (number)|14]], [[34 (number)|34]], [[82 (number)|82]], 198, [[478 (number)|478]], …

Like the relationship between [[Fibonacci number]]s and [[Lucas number]]s,
:<math>Q_n=\frac{P_{2n}}{P_n}</math>
for all [[natural number]]s ''n''.

The companion Pell numbers can be expressed by the closed form formula
:<math>Q_n=\left(1+\sqrt 2\right)^n+\left(1-\sqrt 2\right)^n.</math>

These numbers are all [[parity (mathematics)|even]]; each such number is twice the numerator in one of the rational approximations to <math>\sqrt 2</math> discussed above.

Like the Lucas sequence, if a Pell–Lucas number {{sfrac|1|2}}''Q<sub>n</sub>'' is prime, it is necessary that ''n'' be either prime or a [[power of 2]]. The Pell–Lucas primes are
:3, 7, 17, 41, 239, 577, … {{OEIS|id=A086395}}.

For these ''n'' are 
:2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, … {{OEIS|id=A099088}}.