Non-homogeneous conditional recurrences

A non-homogeneous conditional recurrence sequence (Formula presented.) is one in which the linear recurrence satisfied by (Formula presented.) depends on the residue of n modulo some integer (Formula presented.) and at least one of these linear recurrences is non-homogeneous. We describe a new linear algebraic method, using the successor operator and the annihilator matrix, to get a linear recurrence satisfied by the sequence