Conjectured dominion formula for cycles

Let $G$ be a cycle on $n\ge 3$ vertices. Its dominion is the invariant denoted by $\zeta(G)$. Cycle dominion conjecture. If $G$ is a cycle on $n\ge 3$ vertices, then

$$\zeta(G)=\begin{cases} \frac{n^2+5n}{6}, & \text{if } n \equiv 1\pmod3 \\ n, & \text{if } n \equiv 2\pmod3. \end{cases}$$

The cases $n\equiv 1,2\pmod3$ are presented as conjectural because the corresponding values are extrapolated from early recurring values using Lagrange interpolation; the case $n\equiv 0\pmod3$ was established separately in the preceding argument.