To be specific, there have been questions on this topic:
This wiki contains spoliers about these questions, so you are recommended to do them first before reading the rest of this note.
In these questions: Actually, please do those questions first.
In these questions, \(k=2\) and \(n = 0\) to \(n=5\) and \(n=0\) to \(n=6\).
There's a lot of similarities, and even the answer is amazing, but I shall not spoil it.
Both questions are in the form "\(f(n)\) is a \(p\)-degree polynomial. When \(q = 0,1,2,\ldots,p\), \( f(q) = k^q\). What is \(f(x)\)?"
These questions are a special case of this form, with \(k = 2\) and x = 2*q+1. The answer is also very similar, f(x) = 2^(2*q).
I would like to know if this can be generalised for q and also, k, and if so, what values of x can create these values.
I'll work on it when I have time. Which doesn't come a lot.