Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T19:38:51.669Z Has data issue: false hasContentIssue false
  • English
  • Français

A VARIANT OF HOFSTADTER’S SEQUENCE AND FINITE AUTOMATA

Part of: Sequences and sets Elementary number theory

Published online by Cambridge University Press:  03 May 2013

JEAN-PAUL ALLOUCHE  and
JEFFREY SHALLIT
JEAN-PAUL ALLOUCHE
Affiliation:
CNRS, Institut de Mathématiques, Université Pierre et Marie Curie, Case 247, 4 place Jussieu, F-75752 Paris Cedex 05, France email allouche@math.jussieu.fr
JEFFREY SHALLIT*
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
*
✉shallit@cs.uwaterloo.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Following up on a paper of Balamohan et al. [‘On the behavior of a variant of Hofstadter’s $q$-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given.

Keywords

Hofstadter’s sequencerecurrenceautomatic sequencefinite automaton

MSC classification

Secondary: 11B85: Automata sequences 11B37: Recurrences 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11A63: Radix representation; digital problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 1-2 , October 2012 , pp. 1 - 8
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

References

Allouche, G., Allouche, J.-P. and Shallit, J., ‘Kolam indiens, dessins sur le sable aux îles Vanuatu, courbe de Sierpinski et morphismes de monoïde’, Ann. Inst. Fourier 56 (2006), 2115–2130.CrossRefGoogle Scholar
Allouche, J.-P. and Shallit, J., Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Allouche, J.-P. and Shallit, J., ‘The ring of $k$-regular sequences, II’, Theoret. Comput. Sci. 307 (2003), 3–29.CrossRefGoogle Scholar
Balamohan, B., Kuznetsov, A. and Tanny, S., ‘On the behavior of a variant of Hofstadter’s$q$-sequence’, J. Integer Seq. 10 (2007), (Article 07.7.1).Google Scholar
Dalton, B., Rahman, M. and Tanny, S., ‘Spot-based generations for meta-Fibonacci sequences’, Exp. Math. 20 (2011), 129–137.CrossRefGoogle Scholar
Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T., Recurrence Sequences (American Mathematical Society, 2003).CrossRefGoogle Scholar
Hofstadter, D., Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, New York, 1979).Google Scholar
Isgur, A., Rahman, M. and Tanny, S., ‘Solving nonhomogeneous nested recursions using trees’, Preprint, May 12 2011, arXiv:1105.2351.Google Scholar
Pinn, K., ‘Order and chaos in Hofstadter’s $q(n)$ sequence’, Complexity 4 (1999), 41–46.3.0.CO;2-3>CrossRefGoogle Scholar
Rahman, M., ‘A combinatorial interpretation of Hofstadter’s $G$-sequence’, Atl. Electron. J. Math. 5 (2012), 16–21.Google Scholar
Sloane, N. J. A., ‘The on-line encyclopedia of integer sequences’, Electronic database available at http://oeis.org.Google Scholar