On a Generalization of Hofstadter’s Q‐Sequence: A Family of Chaotic Generational Structures

Abstract

Hofstadter Q-recurrence is defined by the nested recurrence Qn=Qn−Qn−1+Qn−Qn−2, and there are still many unanswered questions about certain solutions of it. In this paper, a generalization of Hofstadter’s Q-sequence is proposed and selected members of this generalization are investigated based on their chaotic generational structures and Pinn’s statistical technique. Solutions studied have also curious approximate patterns and considerably similar statistical properties with Hofstadter’s famous Q-sequence in terms of growth characteristics of their successive generations. In fact, the family of sequences that this paper introduces suggests the existence of conjectural global properties in order to classify unpredictable solutions to Q-recurrence and a generalization of it.

On a conjecture about generalized Q-recurrence

“…In this section, an existence of breaking point for termination of Q d, (n) is investigated. It is a relatively easy fact that Q , (n) is a nite sequence due to Q , ( ) = [21]. On the other hand, our following analysis con rms that the termination of Q , (n) is an exceptional behaviour for Q d, (n), at least, in the range of experiments that the next section focuses on.…”

“…Recently, the chaotic generational structures of certain members of Q d, (n) and Q d, (n) were investigated thanks to known methods in empirical literature [21][22][23] in order to search a conjectural global property for rescaling of amplitudes of successive generations. On the other hand, for ascending values of d and , behaviours of Q d, (n) become much more complicated as well as conserving their interesting properties which are reported in the following sections.…”

“…This sequence is highly chaotic and nite since Q( ) = . On the other hand, the same recurrence can have slow solution with appropriate initial conditions [20] while it can be naturally quasi-periodic for certain set of initial conditions such as A296518 and A296786 in OEIS [21]. Also, intriguing generational structures which are analysed by spot-based generation concept appear with a family of initial conditions for Q(n) = Q(n − Q(n − )) + Q(n − Q(n − )) + Q(n − Q(n − )) [21].…”

“…On the other hand, the same recurrence can have slow solution with appropriate initial conditions [20] while it can be naturally quasi-periodic for certain set of initial conditions such as A296518 and A296786 in OEIS [21]. Also, intriguing generational structures which are analysed by spot-based generation concept appear with a family of initial conditions for Q(n) = Q(n − Q(n − )) + Q(n − Q(n − )) + Q(n − Q(n − )) [21]. This behavioral diversity observed in three-term Hofstadter Q-recurrence can be interpreted as a sign of di culty of problems about certain types of nested recurrences, especially for Hofstadter-like recurrences.…”

“…On the other hand, the coexistence of order and chaos in some unpredictable meta-Fibonacci sequences can be recreated experimentally thanks to inital condition patterns in terms of considerable behavioral similarities. At this point, it would be bene cial to remember the de nition of a generalization of Hofstadter's Q-sequence [21].…”