Number of Hamiltonian cycles on a Sierpiński graph
Nice idea! The problem seems to be in step $(b)$. Replacing $(X_1,B,X_2)$ in an $N$-path by $(X_1,X_2)$ gives an $\bar{N}$-path, but not every $\bar{N}$-path will contain $(X_1,X_2)$. So this is not a bijection. This only says $N(n) \leq \bar{N}(n)$.

Or you can in fact show that $\bar{N}(n) = 3N(n)/2$, resulting in $N(n+1) = 3N^3$.