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

Y.-F. S. Pétermann; J.-L. Rémy; I. Vardi

Journal de théorie des nombres de Bordeaux (1999)

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

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topPétermann, Y.-F. S., Rémy, J.-L., and Vardi, I.. "On a functional-differential equation related to Golomb's self-described sequence." Journal de théorie des nombres de Bordeaux 11.1 (1999): 211-230. <http://eudml.org/doc/248337>.

@article{Pétermann1999,

abstract = {The functional-differential equation $f^\{\prime \}(t) = 1/f ( f(t))$ is closely related to Golomb’s self-described sequence $F$,\begin\{equation*\} \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 .\end\{equation*\}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 $E(n)$ in the asymptotic expression $F(n) = \phi ^\{2-\phi \} n^\{\phi -1\} + E(n)$ (where $\phi $ is the golden number).},

author = {Pétermann, Y.-F. S., Rémy, J.-L., Vardi, I.},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {self-generated sequence; Golomb sequence; functional-differential equation},

language = {eng},

number = {1},

pages = {211-230},

publisher = {Université Bordeaux I},

title = {On a functional-differential equation related to Golomb's self-described sequence},

url = {http://eudml.org/doc/248337},

volume = {11},

year = {1999},

}

TY - JOUR

AU - Pétermann, Y.-F. S.

AU - Rémy, J.-L.

AU - Vardi, I.

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

JO - Journal de théorie des nombres de Bordeaux

PY - 1999

PB - Université Bordeaux I

VL - 11

IS - 1

SP - 211

EP - 230

AB - The functional-differential equation $f^{\prime }(t) = 1/f ( f(t))$ is closely related to Golomb’s self-described sequence $F$,\begin{equation*} \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 .\end{equation*}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 $E(n)$ in the asymptotic expression $F(n) = \phi ^{2-\phi } n^{\phi -1} + E(n)$ (where $\phi $ is the golden number).

LA - eng

KW - self-generated sequence; Golomb sequence; functional-differential equation

UR - http://eudml.org/doc/248337

ER -

## References

top- [Fi] N.J. Fine.Solution to problem 5407. Amer. Math. Monthly74 (1967), 740-743. MR1534405
- [Go] S.W. Golomb.Problem 5407. Amer. Math. Monthly73 (1966), 674.
- [Ma] Daniel Marcus.Solution to problem 5407. Amer. Math. Monthly74 (1967), 740. MR1534405
- [McK] M.A. McKiernan.The functional differential equation D f = 1/ff. Proc. Amer. Math. Soc.8 (1957), 230-233. Zbl0078.11905MR84096
- [Pé] Pétermann Y.-F.S.On Golomb's self describing sequence II. Arch. Math.67 (1996), 473-477. Zbl0865.11022MR1418909
- [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. Zbl0901.11009MR1631240
- [Ré] Jean-Luc Rémy.Sur la suite autoconstruite de Golomb. J. Number Theory66 (1997), 1-28. Zbl0881.11024MR1467187
- [Va] Ilan Vardi.The error term in Golomb's sequence. J. Number Theory40 (1992), 1-11. Zbl0758.11012MR1145850

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