A Chaotic Cousin Of Conway's Recursive Sequence

Abstract

I study the recurrence D(n)= D(D(n-1))+D(n-1-D(n-2)), D(1)=D(2)=1. Its definition has some similarity to that of Conway's sequence defined through a(n)= a(a(n-1))+a(n-a(n-1)), a(1)=a(2)=1. However, in contradistinction to the completely regular and predictable behaviour of a(n), the D-numbers exhibit chaotic patterns. In its statistical properties, the D-sequence shows striking similarities with Hofstadter's Q(n)-sequence, defined through Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2)), Q(1)=Q(2)=1. Compared to the Hofstadter sequence, the D-recurrence shows higher structural order. It is organized in well-defined ``generations'', separated by smooth and predictable regions. The article is complemented by a study of two further recurrence relations with definitions similar to those of the Q-numbers. There is some evidence that the different sequences studied share a universality class. Could it be that there are some real life processes modelled by these recurrences? I OFFER A CASH PRIZE OF $100 TO THE FIRST PROVIDING A PROOF OF SOME CONJECTURES ABOUT D(n) FORMULATED IN THIS ARTICLE.