I was working on the diophantine equation $7x^2+y^2=2^n$ for each natural $n\geq 3$. I proved by induction that $(x_n,y_n)$ is an integer solution for $n\geq 3,$ where $x_3=y_3=1$ and $$x_{n+1}=\frac{x_n+y_n}{2} $$ $$y_{n+1}=\frac{7x_n-y_n}{2} $$ So I obtained a family of infinite solutions for each $n$ given by a system of recurrence sequences.
I wonder if there is a general theory about this, which relate diophantine equations with recurrence sequences.
Thank you in advance.
In the ring $\mathbb Q[\sqrt{-7}]$ the algebraic integers are of the form $\frac a2+\frac b2\sqrt{-7}$ where $a,b$ are integers of the same parity (both odd or both even.)
It’s a bit long to describe in an answer, but your $(x_n,y_n)$ can be seen as $$\frac{x_n}2+\frac{y_n}2\sqrt{-7}=\left(\frac12+\frac12\sqrt{-7}\right)^{n-2}$$
More generally, if:
$$\frac x2+\frac y2\sqrt{-7}=\left(\frac a2+\frac b2\sqrt{-7}\right)^k$$ then $\frac{x^2+7y^2}4=\left(\frac{a^2+7b^2}4\right)^k.$
So you can solve $x^2+7y^2=11^k$ using $a=4,b=2.$
This kind of argument works for any $\sqrt{D},$ where $D$ is square-free. If $D\equiv 1\pmod 4$ you have to use the half-integers, like for $-7,$ but for other $D$ you only are allowed to take $a+b\sqrt D$ where $a,b$ are integers.
Study Quadratic Fields.