When does a polynomial $f(x)$ generate an arithmetic sequence for consecutive values of integer $x$?

Question

Given polynomial $f(x) = a_0 + a_1x + \dots + a_n x^n, a_n \ne 0, n \ge 2, k \in (0,n), a_k \in \{0, 1, 2, \dots\}$, under what conditions is $f(r), f(r+1), \dots f(r+n)$ an arithmetic sequence for integer $r$?

When does $f(x)$ not generate an $(n+1)$ term arithmetic sequence? (Update: Never - based on responses to this question).

Updated question: When does $f(x)$ generate an $n$ term arithmetic sequence? Given n values, we can fit an $n$-degree polynomial. The question is given the polynomial, when does it generate an $n$-term arithmetic sequence for $n$ consecutive values of $x$

Note that $a_k$ are fixed in this problem.

Answer 1

In one direction, this obviously happens for every $r$ when $f(x)$ is linear. In the other direction, suppose $f(x)$ is not linear, then we can write the $n+1$ values $f(r), f(r+1), ..., f(r+n)$ as $a + bx$ for $x = r, r+1, ..., r+n$. Then the polynomial $$g(x) = f(x) - (a + bx)$$ is a nonzero polynomial of degree $n$ but has $n+1$ zeroes, which is impossible by the Fundamental Theorem of Algebra. So, such an arithmetic sequence exists if and only if $f(x)$ is linear.

Answer 2

If the sequence has $n$ terms, one can fit a $n + 1$ or higher degree polinomial that agrees on that points (and has arbitrary values at other, selected points).