Does anyone know if the following problem has ever been studied?

Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$ where $n$ is a positive integer.

Does there exist a value of $k$ such that if $p_n(a,b)$ is an integer for $k$ consecutive values of $n$ then $p_n(a,b)$ is an integer for every $n$? If so what is the minimum value of $k$?

What happens if we change the 'integer' condition to 'rational' values at the previous question?

It's not difficult to establish some recurrence relation among the values of $p_n$ but none of them seem to be promising.

I would like to know any reference for this problem or how it could be solved.

Any help would be appreciated.

A quotient of the ring of symmetric functions generalizing quantum cohomology, since the $p_n$ are just the complete homogeneous symmetric polynomials $h_n$ in the two variables $a$ and $b$. But the "rational" version doesn't follow from this argument. $\endgroup$ – darij grinberg Nov 17 '18 at 16:21