Morphic Words and Nested Recurrence Relations

TL;DR

This paper investigates nested recurrence relations linked to morphisms over infinite alphabets, analyzing their properties and connections to famous sequences, and establishing conditions for their asymptotic behavior.

Contribution

It introduces a framework for understanding nested recurrences via morphisms and characterizes the limits of their normalized sequences.

Findings

Connections to Hofstadter's G, Conolly, and Tanny sequences

Necessary and sufficient conditions for the limit a(n)/n to exist

Analysis of recurrences with finitely many terms

Abstract

We explore a family of nested recurrence relations with arbitrary levels of nesting, which have an interpretation in terms of fixed points of morphisms over a countably infinite alphabet. Recurrences in this family are related to a number of well-known sequences, including Hofstadter's G sequence and the Conolly and Tanny sequences. For a recurrence a(n) in this family with only finitely terms, we provide necessary and sufficient conditions for the limit a(n)/n to exist.