MathNet 1988 — Part 01 Question 9

Let $x_{n+1} = 4 x_n - x_{n-1}$, $x_0 = 0$, $x_1 = 1$, and $y_{n+1} = 4 y_n - y_{n-1}$, $y_0 = 1$, $y_1 = 2$.
Show for all $n \geq 0$ that $y_n^2 = 3 x_n^2 + 1$.