Vanishing second differences force an affine sequence

Let $R$ be a commutative ring, let $e$ be a natural number, and let $a \colon \mathbb{N} \to R$ be a sequence. Assume that the second differences of $a$ vanish in the range determined by $e$, namely that $a_k - 2a_{k+1} + a_{k+2} = 0$ for every natural number $k$ with $k + 1 < e$; note that this hypothesis is vacuous when $e \le 1$. Then for every natural number $k$ with $k \le e$ one has

where $k$ is understood via the canonical ring homomorphism $\mathbb{N} \to R$. Thus on the initial segment $\{0, 1, \dots, e\}$ the sequence is the arithmetic progression with initial term $a_0$ and common difference $a_1 - a_0$. The recurrence is stated in the shifted form with indices $k$, $k+1$, $k+2$ rather than $k-1$, $k$, $k+1$, so that no truncated subtraction on $\mathbb{N}$ occurs; the conclusion is asserted for all indices up to and including $e$, one step beyond the last index at which the recurrence is assumed.

This is the elementary statement that solutions of the linear recurrence $a_{k+2} = 2a_{k+1} - a_k$, whose characteristic polynomial is $(x-1)^2$, are exactly the affine sequences — equivalently, that a discrete harmonic function on a path is affine. It is used in the analysis of divisors supported on the special fibre of a resolution of an $A_{e-1}$ singularity, where the condition of zero intersection with each exceptional component is precisely the vanishing of second differences; the consumer is MvPolynomial.CrossingQuotient.Resolution.exists_open_pullback_twist_iso_tensorUnit_of_degree_eq_zero.