Lisansüstü Eğitim Enstitüsühttps://hdl.handle.net/20.500.12960/102024-10-03T22:42:16Z2024-10-03T22:42:16ZOn Hofstadter Heart SequencesAlkan, AltuğFox, NathanAybar, Orhan Özgürhttps://hdl.handle.net/20.500.12960/4122021-06-05T19:56:59Z2017-01-01T00:00:00ZOn Hofstadter Heart Sequences
Alkan, Altuğ; Fox, Nathan; Aybar, Orhan Özgür
The Hofstadter Q-sequence and the Hofstadter-Conway $10000 sequence are perhaps the two best known examples of meta-Fibonacci sequences. In this paper, we explore an unexpected connection between them. When the Q-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 Q-sequence and illustrate how it also generates heart patterns when subtracted from the Conway sequence.
0000-0001-8353-2106; 0000-0001-8353-2106; WOS:000415970800001
2017-01-01T00:00:00ZAn exploration of solutions to two related Hofstadter-Huber recurrence relationsAlkan, AltuğFox, NathanAybar, Orhan ÖzgürAkdeniz, Zehrahttps://hdl.handle.net/20.500.12960/4112021-06-05T19:56:58Z2020-01-01T00:00:00ZAn exploration of solutions to two related Hofstadter-Huber recurrence relations
Alkan, Altuğ; Fox, Nathan; Aybar, Orhan Özgür; Akdeniz, Zehra
In this study, we explore the properties of certain solutions of two Hofstadter-Huber recurrence relations. The first is Hofstadter's V-recurrence, which is defined by the nested recurrence relation V(n) = V(n - V (n - 1)) + V (n - V (n - 4)) . Plus, we introduce another meta-Fibonacci recurrence H(n) = H(n - H(n - 2)) + H(n - H(n - 3)) . First, we study a finite chaotic solution to the V-recurrence in order to analyse its generational structure. Then, we explore a new type of infinite solution to nested recurrence relations, finding solutions of this type to both the V-recurrence and the H-recurrence. Our construction relates to systems of nested recurrences that resemble Golomb's recurrence G(n) = G(n - G(n - 1)) + 1 . (C) 2020 Elsevier Ltd. All rights reserved.
WOS:000571059300007
2020-01-01T00:00:00ZOn a conjecture about generalized Q-recurrenceAlkan, Altuğhttps://hdl.handle.net/20.500.12960/4092021-06-05T19:56:58Z2018-01-01T00:00:00ZOn a conjecture about generalized Q-recurrence
Alkan, Altuğ
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.
WOS:000456406700001
2018-01-01T00:00:00ZOn a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational StructuresAlkan, Altuğhttps://hdl.handle.net/20.500.12960/4102021-06-05T19:56:58Z2018-01-01T00:00:00ZOn a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures
Alkan, Altuğ
Hofstadter Q-recurrence is defined by the nested recurrence Q(n) = Q(n - Q(n - 1)) + Q(n - Q(n - 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.
WOS:000437967700001
2018-01-01T00:00:00Z