The odd-order exact formula for replication numbers of paths

Let $P_n$ be the path on $n$ vertices, and let $\rho_R(P_n)$ denote its replication number. For odd $n\geq 1$, write the upper-bound formula according to the residue class of $n$ modulo $4$. The odd-order exact formula. The upper bound is tight, namely

$$\rho_R(P_n)=\begin{cases} n^2/4+n^2/16+3n/8+5/16,& n\equiv 1\pmod 4,\\ n^2/4+n^2/16+3n/8+1/16,& n\equiv 3\pmod 4. \end{cases}$$

Computer searches determine the exact values for $n\leq 16$, and this conjecture proposes the displayed formula for all odd $n$.