 Sawtooth Ising chain

An Ising model on a chain of corner-sharing triangles, with one base spin and one apex spin per cell. If the apex interacts only with the two base corners, summing each apex independently gives a two-by-two transfer matrix on base spins. For symmetric Ising spins and energy $Js{s}^{\prime }+\bar{J}\bar{s}(s+s^{\prime })$, its entries are $2e^{-\beta Js{s}^{\prime }}\cosh [\beta \bar{J}(s+s^{\prime })]$.