Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.
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