KöMaL 2024 A880

Find all triples $(a,b,c)$ of real numbers for which there exists a function $f\colon \mathbb{Z}^+ \to \mathbb{Z}^+$ satisfying $af(n)+bf(n+1)+cf(n+2)<0$ for every $n \in \mathbb{Z}^+$ ($\mathbb{Z}^{+}$ denotes the set of positive integers).