Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series

We study a period-two family of adjacent-sum lattice polytopes whose consecutive-coordinate bounds alternate between $s$ and $s+1$. This provides a simple non-uniform deformation of the classical uniform model while retaining an explicit transfer-matrix structure. The lattice-point counts exhibit a parity split: the odd- and even-dimensional sequences have distinct rational generating functions with a common denominator. The odd-dimensional series satisfies a M\"obius recurrence and admits an arctangent closed form, whereas the even-dimensional series obeys a coupled recurrence. Their common dominant pole determines the exponential growth in both parity classes. For the cyclic model obtained by adding a constraint between the first and last coordinates, the count becomes a matrix trace. The two cyclic parity classes again have rational generating functions with the same denominator; the even-dimensional numerator has a Jacobi-derivative form, while the odd-dimensional one is given by an explicit anti-diagonal cofactor expression. We also derive dimension-generating functions for fixed dilations, linear recurrences for lattice-point counts, rational volume-generating functions, and a bivariate identity for the coefficients of the $h^*$-polynomials. When $s=1$, every even-dimensional polytope decomposes into a Cartesian product of unimodular triangles, yielding explicit formulas and the Gorenstein property. For every $s\geq 2$, the Gorenstein property fails in some even dimension.