A Slow Relative of Hofstadter's Q-Sequence

TL;DR

This paper introduces a new recurrence relation extending Hofstadter's Q-sequence, resulting in a monotonically increasing sequence with predictable frequency properties, and discusses its limitations in generalization.

Contribution

The paper proposes a third-order recurrence relation that produces a monotonic sequence and provides a detailed analysis of its frequency sequence and computational aspects.

Findings

Sequence increases monotonically by 0 or 1

Complete description of the frequency sequence

Sequence cannot be easily generalized

Abstract

Hofstadter's Q-sequence remains an enigma fifty years after its introduction. Initially, the terms of the sequence increase monotonically by 0 or 1 at a time. But, Q(12)=8 while Q(11)=6, and monotonicity fails shortly thereafter. In this paper, we add a third term to Hofstadter's recurrence, giving the recurrence B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3)). We show that this recurrence, along with a suitable initial condition that naturally generalizes Hofstadter's initial condition, generates a sequence whose terms all increase monotonically by 0 or 1 at a time. Furthermore, we give a complete description of the resulting frequency sequence, which allows the nth term of our sequence to be efficiently computed. We conclude by showing that our sequence cannot be easily generalized.