Concrete Math
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It is shown that the number f"@l of free subgroups of index 6@l in the modular group PSL"2(Z), when considered modulo a prime power p^@a with p=5, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing PSL"2(Z) as a special case), and for a one-parameter family of lifts of the Hecke group H(4)=C"2@?C"4. All this is achieved by explicitly determining Pade approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complement previous work by Kauers and the authors (2012) [12,15], where it is shown, among other things, that the free subgroup numbers of PSL"2(Z) and its lifts display rather complex behaviour modulo powers of 2 and 3.