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Increasing Self-Described Sequences
Y.-F.S. Pétermann1 and Jean-Luc Rémy2
1Université de Genève, Section de Mathématiques, 2-4, rue du Liévre, C.P. 240, 1211 Genève 24, Switzerland
Petermann@math.unige.ch
2Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), B.P. 239, 54506 Vandoe uvre-lès-Nancy Cedex, France
Jean-Luc.Remy@loria.fr
Annals of Combinatorics 8 (3) p.325-346 September, 2004
AMS Subject Classification:11B37, 11N37, 34K25
Abstract:
We proceed with our study of increasing self-described sequences F, beginning with 1 and defined by a functional equation and Fi denoting ). In [1] we exhibited the simple solution , for some , of the associated functional-differential equation , and we proved that provided , where , we have the asymptotic equivalence .

    In the present paper we show that this last result is optimal, in the sense that the self-described sequence defined by , that is

1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, ...,
for which the boundary case holds, does not satisfy . We also show that the m-th term F(m) of a sequence F for which the boundary case holds is nevertheless of asymptotic order .

    Then we investigate the behaviour of self-described sequences F when lies beyond the boundary case. In [1] we established the estimates when is the unique fixed point of a certain associated function. We were only able to prove in general that the latter holds when does not lie beyond the boundary case, however. In the present paper we prove that whenever is the unique fixed point of this function, and in addition we obtain estimates more precise than . This applies for instance to the sequence defined by , that is

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, ... .

Keywords: self-described sequences, functional-differential equations

References:

1. Y.-F.S. Pétermann, J.-L. Rémy, and I. Vardi, Discrete derivatives of sequences, Adv. Appl. Math. 27 (2001) 562--584.