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On a functional-differential equation related to Golomb's self-described sequence

Y.-F. S. Pétermann, J.-L. Rémy, I. Vardi (1999)

Journal de théorie des nombres de Bordeaux

The functional-differential equation f ' ( t ) = 1 / f ( f ( t ) ) is closely related to Golomb’s self-described sequence F , 1 , 1 , 2 , 2 , 2 , 3 , 3 , 2 , 4 , 4 , 4 3 , 5 , 5 , 5 , 3 , 6 , 6 , 6 , 6 , 4 , . We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number p 0 there is exactly one increasing solution with p as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely many times. In fact...

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