We have the generating function $$1/(1−2x−x^2)=\sum_{n=0}^{\infty}a_nx^n$$ Show $$a_n^2 + a_{n+1}^2 = a_{2n+2}$$.
I am thinking of finding a 2 × 2 matrix A and the text gives a hint of $$A^{n+2}= \left( \begin{smallmatrix} a_n&a_{n+1}\\ a_{n+1}&a_{n+2} \end{smallmatrix} \right)$$ and consider the top left entry of the matrix product $A^{n+2}A^{n+2}$. I do not exactly understand this hint.
Hint:This may help you to solve the problem:
$$\frac{1}{1-2x-x^2}=\frac{1}{2\sqrt2}.(\frac{1}{x+1-\sqrt2}-\frac{1}{x+1+\sqrt 2}$$
Now expand $\frac{1}{1+(x-\sqrt2)}$ and $\frac{1}{1+(x+\sqrt2)}$ and sum up terms with identical powers, you get the expansion of $\frac{1}{1-2x-x^2}$. Now you can check $a^2_n+a^2_{n+1}= a_{2n+2}$.