General conditional recurrences

A general conditional recurrence sequence {qn}{qn} is one in which the recurrence satisfied by qnqn depends on the residue of n   modulo some integer r⩾2r⩾2. The properties of such sequences are studied, and in particular it is shown that any such sequence {qn}{qn} satisfies a single recurrence equation not dependent on the modulus r. We also obtain generating functions and Binet-like formulas for such sequences.