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

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

top Access to full text Full (PDF) Access to full text

## 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).

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.