Ground-state counting of the frustrated sawtooth Ising chain (${\Omega }_L=3^L+(-1{)}^L$)

For positive equal exchange constants and $2L$ symmetric Ising spins on a ring, a minimum-energy triangle has precisely one spin different from its other two. Given adjacent base spins, there is one permitted apex choice if they agree and two if they disagree. The counting transfer matrix is $C=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$, with eigenvalues $3,-1$, giving ${\Omega }_L=\mathrm{Tr}{C}^L=3^L+(-1{)}^L$. Thus the residual entropy per spin is $(k_B/2)\log 3$. If the two apex bonds are strictly stronger than the base bond, only the two globally reversed configurations minimize all triangles and this residual entropy vanishes.