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

Y.-F.-S. Pétermann Jean-Luc Rémy 1 I. Vardi
1 POLKA - Polynomials, Combinatorics, Arithmetic
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : The functional-differential equation $f'(t)=1/f(f(t))$ is closely related to Golomb's self-described sequence, denoted F. We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for eveery 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 we conjecture that the difference of two increasing solutions behave very similarly as the error term $E(n)$ in the asymptotic expression $F(n) = \phi^{2-\phi}n^{\phi-1} + E(n)$(where $\phi$ is the golden number). || l'équation différentielle-fonctionnelle $f'(t)=1/f(f(t))$ a des liens étroits avec la suite auto-décrite de Golomb, notée $F$. Nous décrivvons les solutions croissantes de cette équation. Nous montrons qu'une telle solution possède néces
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Submitted on : Tuesday, September 26, 2006 - 8:39:13 AM
Last modification on : Friday, February 4, 2022 - 3:30:17 AM


  • HAL Id : inria-00098846, version 1



Y.-F.-S. Pétermann, Jean-Luc Rémy, I. Vardi. On a functional-differential equation related to Golomb's self-described sequence. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 1999, 11, pp.211-230. ⟨inria-00098846⟩



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