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In mathematics, the Pell numbers and Companion Pell Numbers (Pell-Lucas Numbers) are both sequences of integers.
Pell numbers
The Pell numbers are defined recursively by:
- <math>
F(n)=
\left\{
\begin{matrix}
0\,,\qquad\qquad\qquad\quad\,\ \ \,&&\mbox{if }n=0\,;\ \ \\
1,\qquad\qquad\qquad\qquad\,&&\mbox{if }n=1;\ \ \,\\
2F(n-1)+F(n-2)&&\mbox{otherwise.}
\end{matrix}
\right.
<math>
In words: you start with 0 and 1, and then produce the next Pell number by adding twice the previous Pell number to the Pell number before that. The first few terms of the sequence are 0, 1, 2, 5, 12, 29, 70, 169, 408...
Companion Pell Numbers (Pell-Lucas Numbers)
The Companion Pell numbers are defined recursively by:
- <math>
F(n)=
\left\{
\begin{matrix}
2\,,\qquad\qquad\qquad\quad\,\ \ \,&&\mbox{if }n=0\,;\ \ \\
2,\qquad\qquad\qquad\qquad\,&&\mbox{if }n=1;\ \ \,\\
2F(n-1)+F(n-2)&&\mbox{otherwise.}
\end{matrix}
\right.
<math>
In words: you start with 2 and 2, and then produce the next Pell number by adding twice the previous Pell number to the Pell number before that. The first few terms of the sequence are 2, 2, 6, 14, 34, 82, 198, 478...
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