On a functional-differential equation related to Golomb's self-described sequence
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.
scientific article; zbMATH DE number 1370113