Pell Number as Sum of Squares

Theorem
Let $P_n$ be a Pell Number:


 * $P_n = \begin{cases} 0 & : n = 0 \\

1 & : n = 1 \\ 2 P_{n - 1} + P_{n - 2} & : \text {otherwise}\end{cases}$

Then:
 * $P_{2 n + 1} = P_{n + 1}^2 + P_n^2$

Proof
This proof proceeds by induction.

Basis for the Induction
$\map P 0$ is the case:

So $\map P 0$ is seen to hold.

$\map P 1$ is the case:

So $\map P 1$ is seen to hold.

Induction Hypothesis
Now we need to show that, if $\map P {n - 1}$ and $\map P n$ are true, where $n \ge 1$, then it logically follows that $\map P {n + 1}$ is true.

So this is our induction hypotheses:
 * $P_{2 n - 1} = P_n^2 + P_{n - 1}^2$
 * $P_{2 n + 1} = P_{n + 1}^2 + P_n^2$

from which we are to show:
 * $P_{2 n + 3} = P_{n + 2}^2 + P_{n + 1}^2$

Induction Step
This is our induction step:

The result follows by induction.