Question 80
Let $\left(x_n\right)_{n\in\mathbb N}$ be a real sequence such that $x_0=0$, $x_1=1$, and $x_{n+2}=3x_{n+1}-2x_n$ for every $n\geqslant0$. Also set $y_n=x_n^2+2^{n+2}$ for every natural number $n$. Prove that for every integer $n\gt 0$, $y_n$ is the square of an odd integer.