The Ultimate Pell number - American History Information Guide and Reference

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:

$F(n)= \left\{ \begin{matrix} 0\,,\qquad\qquad\qquad\quad\,\ \ \,&&\mbox{if }n=0\,;\ \ \\ 1,\qquad\qquad\qquad\qquad\,&&\mbox{if }n=1;\ \ \,\\ 2F(n-1)+F(n-2)&&\mbox{otherwise.} \end{matrix} \right.$

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:

$F(n)= \left\{ \begin{matrix} 2\,,\qquad\qquad\qquad\quad\,\ \ \,&&\mbox{if }n=0\,;\ \ \\ 2,\qquad\qquad\qquad\qquad\,&&\mbox{if }n=1;\ \ \,\\ 2F(n-1)+F(n-2)&&\mbox{otherwise.} \end{matrix} \right.$

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 ...

Categories: Integer sequences

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Pell number

Pell numbers

Companion Pell Numbers (Pell-Lucas Numbers)

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