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Complexity
Volume 2017 (2017), Article ID 2614163, 8 pages
https://doi.org/10.1155/2017/2614163
Research Article

On Hofstadter Heart Sequences

1Graduate School of Science and Engineering, Piri Reis University, Tuzla, 34940 Istanbul, Turkey
2Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ, USA

Correspondence should be addressed to Altug Alkan

Received 14 June 2017; Revised 19 October 2017; Accepted 26 October 2017; Published 20 November 2017

Academic Editor: Peter Giesl

Copyright © 2017 Altug Alkan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Hofstadter -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 -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.