On the number of subgroups of given index in the modular group

A new recurrence for the number of subgroups of given index in the modular group is derived. The proof requires the derivation of a recurrence for a sequence anbn from recurrences for the an and bn. We show that this is always possible if the an and bn satisfy polynomial recurrences. We also include a short proof of a result of W. W. Stothers on the parity of the number of subgroups of given index in the modular group.