If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is

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

In a previous post I asked if $ p_n(a,b)$ was a rational number (or an integer) for $ k$ consecutive values of $ n$ then $ p_n(a,b)$ was a rational number (or an integer) for every $ n$ .

The answer is positive in both cases. If $ p_n(a,b)$ is a rational for 4 consecutive values of $ n$ then $ p_n(a,b)$ is a rational for every $ n$ by the nice answer of Vlad Matei in the previous post, and if $ p_n(a,b)$ is an integer for 4 consecutive values of $ n$ then $ p_n(a,b)$ is an integer for every $ n$ , due to a problem from AMM, namely Problem E2998 by Clark Kimberling.

Do these two results remain true if instead of 4 we have only 3 consecutive values of $ n$ such that $ p_n(a,b)$ is a rational number (or an integer)?

Any help or reference would be appreciated.