On Hofstadter Heart Sequences

Abstract

The Hofstadter -sequence and the Hofstadter-Conway $10000 sequence are perhaps the two best known examples of metaFibonacci sequences. In this paper, we explore an unexpected connection between them. When the -sequence is subtracted from the Conway sequence, a chaotic pattern of heart-shaped figures emerges. We use techniques of Pinn and Tanny et al. to explore this sequence. Then, we introduce and analyze an apparent relative of the -sequence and illustrate how it also generates heart patterns when subtracted from the Conway sequence.

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

“…If the answer is yes, there can be a collection of curious chaotic patterns and generational structures hidden in genes of the Q-recurrence. Solutions with three initial conditions are studied before [17]. In order to go further, computer experiments can be made with four and five initial conditions but empirical results suggest that there is no sign of any sequence family with generational common characteristics except Q-sequence and "Brother" sequence (A284644) although there are some more chaotic solutions such as A278056 [9,11].…”

“…Among the solutions which have an erratic nature, certain variants have underlying structures that contain conjecturally interesting approximate properties such as scaling, self-similarity, and period doubling [15,16]. For these kinds of solutions of nested recurrences, known mathematical techniques for solving difference equations do not work because of the nature of nesting although there are alternative definitions for the generational structure of a chaotic meta-Fibonacci sequence [15][16][17][18][19]. The existence of universality classes for chaotic meta-Fibonacci sequences determined by common characteristics of their respective generational structures is a mysterious open question although there are a variety of attempts in order to search an affirmative answer for this question, partially [15][16][17].…”

“…For these kinds of solutions of nested recurrences, known mathematical techniques for solving difference equations do not work because of the nature of nesting although there are alternative definitions for the generational structure of a chaotic meta-Fibonacci sequence [15][16][17][18][19]. The existence of universality classes for chaotic meta-Fibonacci sequences determined by common characteristics of their respective generational structures is a mysterious open question although there are a variety of attempts in order to search an affirmative answer for this question, partially [15][16][17]. Since the certain solutions to the Hofstadter Q-recurrence are investigated in this study, it would be nice to remember the scatterplots of Hofstadter's original Q-sequence and the "Brother" sequence (A284644) that is defined by Q b n = Q b n − Q b n − 1 + Q b n − Q b n − 2 and initial values Q b 1 = Q b 2 = 2 and Q b 3 = 1 (see Figure 1).…”

“…Since the certain solutions to the Hofstadter Q-recurrence are investigated in this study, it would be nice to remember the scatterplots of Hofstadter's original Q-sequence and the "Brother" sequence (A284644) that is defined by Q b n = Q b n − Q b n − 1 + Q b n − Q b n − 2 and initial values Q b 1 = Q b 2 = 2 and Q b 3 = 1 (see Figure 1). Although there are many solutions to the Hofstadter Q-recurrence with different initial conditions [9, 11-13, 17, 20], these two solutions and their connection based on their generational structures provide a variety of interesting experimental results [17].…”

On a Family of Sequences Related to Prime Counting Function

“…The Hofstadter-Conway $10000 sequence is recursively defined by the nested recurrence relation c(n) = c(c(n − 1)) + c(n − c(n − 1)) and initial values c(1) = c(2) = 1 (Mallows, 1991). This sequence has many amazing properties and a very intriguing generational structure (Alkan, Fox, & Aybar, 2017). One of the most important reasons behind its fascinating nature is the construction of parent spots which are c(n − 1) and n − c(n − 1) and the resulting symmetry that comes from here (Kubo & Vakil, 1996).…”