Golomb's Self Described Sequence and Functional Differential Equations
Identifieur interne : 00C192 ( Main/Merge ); précédent : 00C191; suivant : 00C193Golomb's Self Described Sequence and Functional Differential Equations
Auteurs : Y.-F.-S. Pétermann ; Jean-Luc RémySource :
- Illinois Journal of Mathematics ; 1997.
English descriptors
Abstract
A sequence (word) W of positive integers is {\it self-described}or {\it self-generating} if \tau(W)=W, where \tau(W) is the sequence consisting of the numbers of consecutive equal entries of W. A famous self-generating bounded is Kolakoski's \underbrace{1,}_{1,}\inderbrace{2,}_{2,} \\underbrace{2,}_{1,}\underbrace{1,}_{1,}\\underbrace{2,2,}_{2,}\cdots. In this paper we consider Golomb's sequence F, wich is the only nondecreasing self-generating sequence taking all positive integral values, \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. Let \phi denote the golden number. We prove that F(n)=\phi^{2-\phi}n^{\phi-1}+\frac{n^{\phi-1}}{\log n}h\left(\frac{\log\log n}{\log\phi}\right )+O\left(\frac{n^{\phi-1}{\log^2 n}\log\log n \right), where the real functionh is continuous and satisfies h(x)=-h(x+1) (x\ge0). The method of proof is intimately connected with the more general problem of characterising the solution E_1 of an approximate functional integral equation of the type E_1(t)=-\phi^{1-\phi}t^{\phi-2}\int_2^{\phi}t^{\phi-1}}E_1(u)du+O\left(\frac {t^{\phi-1}}{\log^2}\right, which we discuss in the second part of the paper.
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