Fix an integer $k\geq3$. Define the two families of sequences $\{x_n\}$ and $\{y_n\}$ according to the rules: $$x_n=\frac{x_{n-1}^2+x_{n-2}^2+\cdots+x_{n-k+1}^2}{x_{n-k}} \qquad n\geq k$$ and $$y_n=\frac{(y_{n-1}+y_{n-2}+\cdots+y_{n-k+1})^2}{y_{n-k}} \qquad n\geq k$$ with initial conditions $x_j=y_j=1$ for $j=0,1,\dots,k-1$.

QUESTIONS.(1) It seems that both $x_n$ and $y_n$ are always positive integers.

(2) It also appears true that $y_n=x_n^2$ for all $n$.

Any ideas of a proof? Of course, if (2) holds and $x_n\in\mathbb{N}$ then $y_n\in\mathbb{N}$.

**REMARK.** Special cases are also appreciated, say for $k=3$ etc.