On a functional-differential equation related to Golomb's self-described sequence
type: Item
instance of: scholarly article
title: On a functional-differential equation related to Golomb's self-described sequence
zbMATH Open document ID: 0973.11029
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2001-12-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93732
full work available at URL: jtnb11-1.html
review text: Call self-described, or self-generated any sequence of positive integers that is equal to the sequence of lengths of its consecutive blocks of equal entries. Two classical examples are the Kolakoski sequence \(22112122122\dots\) and the Golomb sequence \(12233 444 555 666.\dots\) The Golomb sequence is the unique nondecreasing self-generating sequence that contains all positive integers. One of the proofs that the \(n\)th term of this sequence is equivalent to \(\varphi^{2-\varphi} n^{\varphi-1}\), where \(\varphi\) is the golden ratio, shows that solutions of the functional-differential equation \(f'(t)= 1/ f(f(t))\) enter the picture. In the paper under review the authors give nice results (and a nice conjecture) on the increasing solutions of this functional-differential equation.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 1370113
zbMATH Keywords: self-generated sequence
zbMATH Keywords: Golomb sequence
zbMATH Keywords: functional-differential equation
author: Y.-F. S. Pétermann
author: J. L. Rémy
author: Ilan Vardi
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
OpenAlex ID: W2057934694
Wikidata QID: Q115158941
cites work: The Functional Differential Equation Df = 1/ff
cites work: On Golomb's self describing sequence. II
cites work: Golomb's self-described sequence and functional differential equations
cites work: On Golomb's self-describing sequence
cites work: The error term in Golomb's sequence
DOI: 10.5802/JTNB.247
description: scientific article; zbMATH DE number 1370113