Given the recursive function $$x_{n+1}=\sqrt{x_n^2+\frac{p}{q}}$$ where $x_0, p, q \in \mathbb{R}$
Can we find values for $x_0$ & $\frac{p}{q}$ so that the array of recursions of $\{x_0, x_1,x_2,x_5,x_{10},...\}$ are rational, such that also the distances between the terms are defined by the $n^{th}$ term of the Fibonnaci series?
For example, if $x_0=0$ and $\frac{p}{q}=1$, the array of $x_n$ that are integers are $\{x_1, x_4, x_9, ..., x_n\}$ where $n$ is a perfect square.
If $x_0=0.5$ and $\frac{p}{q}=2$, then the array of rational numbers are $\{x_1, x_3, x_6, x_{10}, x_{15},..., x_n\}$, where distance between the terms is the $n^{th}$ term of the series of positive integers.
I'll clarify it does not make any sense.