On a functional-differential equation related to Golomb's self-described sequence
Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 211-230.

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).

L’équation différentielle fonctionnelle ${f}^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ a des liens étroits avec la suite auto-décrite $F$ de Golomb,

 $\underset{1,}{\underbrace{1,}}\phantom{\rule{4pt}{0ex}}\underset{2,}{\underbrace{2,2,}}\phantom{\rule{4pt}{0ex}}\underset{2,}{\underbrace{3,3,}}\phantom{\rule{4pt}{0ex}}\underset{3,}{\underbrace{4,4,4}}\phantom{\rule{4pt}{0ex}}\underset{3,}{\underbrace{5,5,5,}}\phantom{\rule{4pt}{0ex}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$
Nous décrivons les solutions croissantes de cette équation. Nous montrons qu’une telle solution possède nécessairement un point fixe non négatif, et que pour chaque nombre $p\ge 0$ il y a exactement une solution croissante ayant $p$ pour point fixe. Nous montrons également qu’en général une condition initiale ne détermine pas une solution unique: les courbes représentatives de deux solutions croissantes distinctes se croisent en effet une infinité de fois. En fait, nous conjecturons que la différence de deux solutions croissantes se comporte de façon très similaire au terme d’erreur $E\left(n\right)$ dans l’expression asymptotique $F\left(n\right)={\phi }^{2-\phi }{n}^{\phi -1}+E\left(n\right)$ (où $\phi$ est le nombre d’or).

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author = {P\'etermann, Y.-F. S. and R\'emy, J.-L. and Vardi, I.},
title = {On a functional-differential equation related to {Golomb's} self-described sequence},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {211--230},
publisher = {Universit\'e Bordeaux I},
volume = {11},
number = {1},
year = {1999},
mrnumber = {1730441},
zbl = {0973.11029},
language = {en},
url = {http://www.numdam.org/item/JTNB_1999__11_1_211_0/}
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Pétermann, Y.-F. S.; Rémy, J.-L.; Vardi, I. On a functional-differential equation related to Golomb's self-described sequence. Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 1, pp. 211-230. http://www.numdam.org/item/JTNB_1999__11_1_211_0/

[Fi] N.J. Fine. Solution to problem 5407. Amer. Math. Monthly 74 (1967), 740-743. | MR

[Go] S.W. Golomb. Problem 5407. Amer. Math. Monthly 73 (1966), 674.

[Ma] Daniel Marcus. Solution to problem 5407. Amer. Math. Monthly 74 (1967), 740. | MR

[McK] M.A. Mckiernan. The functional differential equation D f = 1/ff. Proc. Amer. Math. Soc. 8 (1957), 230-233. | MR | Zbl

[Pé] Pétermann Y.-F.S. On Golomb's self describing sequence II. Arch. Math. 67 (1996), 473-477. | MR | Zbl

[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. | MR | Zbl

[Ré] Jean-Luc Rémy. Sur la suite autoconstruite de Golomb. J. Number Theory 66 (1997), 1-28. | MR | Zbl

[Va] Ilan Vardi. The error term in Golomb's sequence. J. Number Theory 40 (1992), 1-11. | MR | Zbl