Recurrence relation coefficients

Question

For all integers $n \geq 1$, $x_{n+1}$ is linked to $x_n$ by a recurrence relation and $y_{n+1}$ is linked to $y_n$ by another recurrence relation and $x_1$ and $y_1$ are given.

If $A(x_n)^2 + B(x_n y_n) + C(y_n)^2 = 0$ for all integers $n \geq 1$, where $A, B,$ and $C$ are constants, does that mean that $A, B,$ and $C$ all have to be $0$?

Answer 1

It does not.

Suppose that $x_1=2$, $y_1=-2$, $x_{n+1}=2x_n$, $y_{n+1}=2y_n$, $A=C=1$, and $B=2$. The recurrences are easily solved to yield $x_n=2^n$ and $y_n=-2^n$, so $y_n=-x_n$ for all $n\ge 1$, and

$$x_n^2+2x_ny_n+y_n^2=x_n^2-2x_n^2+x_n^2=0$$

for all $n\ge 1$.