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

Pétermann, Y.-F. S., Rémy, J.-L., and Vardi, I.. "On a functional-differential equation related to Golomb's self-described sequence." Journal de théorie des nombres de Bordeaux 11.1 (1999): 211-230. <http://eudml.org/doc/248337>.

@article{Pétermann1999,
abstract = {The functional-differential equation $f^\{\prime \}(t) = 1/f ( f(t))$ is closely related to Golomb’s self-described sequence $F$,\begin\{equation*\} \underbrace\{1,\}\_\{1,\} \ \underbrace\{2,2,\}\_\{2,\} \ \underbrace\{3,3,\}\_\{2,\} \ \underbrace\{4,4,4\}\_\{3,\} \ \underbrace\{5,5,5,\}\_\{3,\} \ \underbrace\{6,6,6,6,\}\_\{4,\} \cdots .\end\{equation*\}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 $p \ge 0$ 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 behaves 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).},
author = {Pétermann, Y.-F. S., Rémy, J.-L., Vardi, I.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {self-generated sequence; Golomb sequence; functional-differential equation},
language = {eng},
number = {1},
pages = {211-230},
publisher = {Université Bordeaux I},
title = {On a functional-differential equation related to Golomb's self-described sequence},
url = {http://eudml.org/doc/248337},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Pétermann, Y.-F. S.
AU - Rémy, J.-L.
AU - Vardi, I.
TI - On a functional-differential equation related to Golomb's self-described sequence
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 1
SP - 211
EP - 230
AB - The functional-differential equation $f^{\prime }(t) = 1/f ( f(t))$ is closely related to Golomb’s self-described sequence $F$,\begin{equation*} \underbrace{1,}_{1,} \ \underbrace{2,2,}_{2,} \ \underbrace{3,3,}_{2,} \ \underbrace{4,4,4}_{3,} \ \underbrace{5,5,5,}_{3,} \ \underbrace{6,6,6,6,}_{4,} \cdots .\end{equation*}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 $p \ge 0$ 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 behaves 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).
LA - eng
KW - self-generated sequence; Golomb sequence; functional-differential equation
UR - http://eudml.org/doc/248337
ER -