Let $C_n$ be an odd (undirected) cycle with $n$ vertices. Let $f$ be the proper 3-coloring of $C_n$ where one vertex is colored $2$ and we alternate $01$ otherwise (0101...012 coloring).
Let $g$ be the proper 3-coloring of $C_n$ where we alternate $012$ as far as we can, and then (in the case $n=1 \mod 3$) fill in appropriately the last vertex so that the coloring remains proper. For example, $C_7$ can be colored $0120121$ ($n=1 \mod 3$), and $C_9$ can be colored $012012012$ ($n=3 \mod 3$). $C_{11}$ can be colored $01201201201$ ($n=2 \mod 3$).
Let $F_k$ be the number of distinct walks on $k$ $f$-colored vertices of $C_n$ ($f(v_1)f(v_2)...f(v_k)$ where $v_1v_2...v_k$ is a walk on $C_n$).
Let $G_k$ be the number of distinct walks on $k$ $g$-colored vertices of $C_n$ ($g(v_1)g(v_2)...g(v_k)$ where $v_1v_2...v_k$ is a walk on $C_n$).
Can we prove that for all $n \geq 7$, $\lim_{k \to \infty} \frac{\log (F_k)}{k} \not = \lim_{k \to \infty} \frac{ \log (G_k)}{k}$?
This is a follow up question to
Counting walks on colored graphs
Thank you.