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We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure and use this to completely describe a class of n-representation-finite algebras called "type A". Contentsdoi:10.1090/s0002-9947-2011-05312-2 fatcat:nxnef3qmijawpo6iu447csa5tq