Consider the cluster algebras $A_n$ and $D_n$. Choose any cluster $x$, is there an explicit formula that express all other cluster variables in terms of $x$?


There is a formula for the Laurent expansions of cluster variables in terms of matchings in "snake graphs" for cluster algebras from any surface. For the question type $A$ (a disk with marked points on the boundary) and type $D$ (a once punctured disk with marked points on the boundary) come from surfaces.

Positivity for cluster algebras from surfaces by Musiker, Schiffler, and Williams has this for any surface. Cluster expansion formulas and perfect matchings by Musiker and Schiffler has treatment for unpunctured surfaces. These notes of Schiffler also treat cluster algebras from surfaces including the Laurent expansion. Canakci and Schiffler also have a series of paper on various aspects of snake graphs.

Edit addition: I originally thought of the more general results for surfaces given above. However, if you are particularly interested in the type $A$ Propp's The combinatorics of frieze patterns and Markoff numbers and Schiffler's A cluster expansion formula (An case) are very nice precursors to the above more general theorems.

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