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We study the family of Y-systems and T-systems associated with thesine-Gordon models and the reduced sine-Gordon models for the parameter ofcontinued fractions with two terms. We formulate these systems by clusteralgebras, which turn out to be of finite type, and prove their periodicitiesand the associated dilogarithm identities which have been conjectured earlier.In particular, this provides new examples of periodicities of seeds.

Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems