Periodica Mathematica Hungarica

, Volume 52, Issue 1, pp 1–17 | Cite as

Remarks on a conjecture on certain integer sequences

  • Shigeki Akiyama
  • Horst Brunotte
  • Attila Pethő
  • Wolfgang Steiner

Summary

The periodicity of sequences of integers <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(a_{n})_{n\in\mathbb Z}$ satisfying the inequalities <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 \ (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.

integer sequences periodicity 

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Copyright information

© Springer-Verlag/Akadémiai Kiadó 2006

Authors and Affiliations

  • Shigeki Akiyama
    • 1
  • Horst Brunotte
    • Attila Pethő
      • 2
    • Wolfgang Steiner
      • 3
    1. 1.Department of Mathematics, Faculty of Science Niigata University
    2. 2.Faculty of Computer Science, University of Debrecen
    3. 3.Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien

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