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

• Volume: 11, Issue: 1, page 211-230
• ISSN: 1246-7405

top

## Abstract

top
The functional-differential equation ${f}^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ is closely related to Golomb’s self-described sequence $F$,$\underset{1,}{\underbrace{1,}}\phantom{\rule{4pt}{0ex}}\underset{2,}{\underbrace{2,2,}}\phantom{\rule{4pt}{0ex}}\underset{2,}{\underbrace{3,3,}}\phantom{\rule{4pt}{0ex}}\underset{3,}{\underbrace{4,4,4}}\phantom{\rule{4pt}{0ex}}\underset{3,}{\underbrace{5,5,5,}}\phantom{\rule{4pt}{0ex}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$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 $E\left(n\right)$ in the asymptotic expression $F\left(n\right)={\phi }^{2-\phi }{n}^{\phi -1}+E\left(n\right)$ (where $\phi$ is the golden number).

## How to cite

top

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 -

## References

top
1. [Fi] N.J. Fine.Solution to problem 5407. Amer. Math. Monthly74 (1967), 740-743. MR1534405
2. [Go] S.W. Golomb.Problem 5407. Amer. Math. Monthly73 (1966), 674.
3. [Ma] Daniel Marcus.Solution to problem 5407. Amer. Math. Monthly74 (1967), 740. MR1534405
4. [McK] M.A. McKiernan.The functional differential equation D f = 1/ff. Proc. Amer. Math. Soc.8 (1957), 230-233. Zbl0078.11905MR84096
5. [Pé] Pétermann Y.-F.S.On Golomb's self describing sequence II. Arch. Math.67 (1996), 473-477. Zbl0865.11022MR1418909
6. [PéRé] Y.-F.S. Pétermann and Jean-Luc Rémy.Golomb's self-described sequence and functional differential equations. Illinois J. Math.42 (1998), 420-440. Zbl0901.11009MR1631240
7. [Ré] Jean-Luc Rémy.Sur la suite autoconstruite de Golomb. J. Number Theory66 (1997), 1-28. Zbl0881.11024MR1467187
8. [Va] Ilan Vardi.The error term in Golomb's sequence. J. Number Theory40 (1992), 1-11. Zbl0758.11012MR1145850

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.