Transitive closure of an affine relation
An answer in case the monoid generated by $A$ for matrix multiplication is finite: Alain Finkel and Jérôme Leroux, How to Compose Presburger-accelerations: Applications to Broadcast Protocols, in FSTTCS 2002 (Foundations of Software Technology and Theoretical Computer Science) 
Lecture Notes in Computer Science 2556, pages 145--156, DOI: 10.1007/3-540-36206-1_14, 2002 (see also the many references there).  The relation $R^k$ is represented by an effectively computable Presburger formula.

A more recent reference on the subject of actually computing the transitive closure is Marius Bozga, Radu Iosif and Filip Konečný, Fast Acceleration of Ultimately Periodic Relations, in CAV 2010 (Computer Aided Verification), Lecture Notes in Computer Science 6174, pages 227--242, DOI: 10.1007/978-3-642-14295-6_23, 2010.