On Hofstadter's G-sequence

TL;DR

This paper explores the properties of Hofstadter's G-sequence, linking it to Wythoff sequences and extending the analysis to other quadratic algebraic numbers, revealing new relationships and proofs.

Contribution

It provides a characterization of Hofstadter's G-sequence via Wythoff sequences and extends the analysis to other quadratic algebraic numbers, offering new proofs and relationships.

Findings

G-sequence characterized by Wythoff sequences

Equality of G-sequence and averages of swapped Wythoff sequences proved

Relationship established between G-sequence and a sequence by Avdivpahic and Zejnulahi

Abstract

We characterize the entries of Hofstadter's G-sequence in terms of the lower and upper Wythoff sequences. This can be used to give a short and comprehensive proof of the equality of Hofstadter's G-sequence and the sequence of averages of the swapped Wythoff sequences. In a second part we give some results that hold when one replaces the golden mean by other quadratic algebraic numbers. In a third part we prove a close relationship between Hofstadter's G-sequence and a sequence studied by Avdivpahic and Zejnulahi.