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.

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.

Chaotic meta-Fibonacci sequences which are generated by intriguing examples of nonlinear recurrences still keep their mystery although substantial progress has been made in terms of well-behaved solutions of nested recurrences. In this study, a recent generalization of Hofstadter’s famous Q-sequence is studied beyond the known methods of generational approaches in order to propose a generalized conjecture regarding the existence of infinitely many different solutions for all corresponding recurrences of this generalization.

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In this study, we introduce a new family of integer sequences which are related to prime-counting function and we focus on some properties of these sequences. Sequence A316434 in OEIS is the fundamental member of solution family that we study. More precisely, we investigate the solutions of recurrence a(n) = a(π(n)) + a(n-π(n)) with some natural initial conditions where π(n) is defined by A000720 in OEIS.