Only $x_0$ is a square. $(x_n,x_{n+1})$ are integer solutions $(x,y)$ of the equation $x^2-4xy+y^2+3=0$ If $x$ is a square, then the equation $y^2-4x^2y+x^4+3=0$ must have integer solutions. $(y-2x^2)^2=3(x^4-1)$ And only $x=1$ is a solution, otherwise must be satisfied $x^2-1=3u^2,\boxed{x^2+1=v^2}$ Simply: This sequence is a $x$ - soluton of the Pell's equation $x^2-3y^2=1$ And $x$ cannot be a square, except $x=1,y=0$

We can use also the characteristical equation method!