Some recurrence relations associated with the Alavi sequence

K. T. Atanassov and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 12, 2006, Number 3, Pages 20—24
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Authors and affiliations

K. T. Atanassov
CBME – Bulgarian Academy of Sciences
Acad. G. Bonchev Str. Block 105, Sofia 1113, Bulgaria

A. G. Shannon
Warrane College, University of New South Wales, NSW 1465, Australia &
KvB Institute of Technology, North Sydney, NSW 2060, Australia

Abstract

This paper considers a modification of the Fibonacci sequence which results in the third order Alavi sequence. Not only are the initial terms quite general but the rule of formation is also modified. Some results are proved to illustrate the underlying structure of the sequence and its relation to known results in the literature. The paper concludes with a suggestion for further research with an arbitrary order extension.

AMS Classification

  • 11B39
  • 11B65

References

    1. Y. Alavi, F.R.K.Chung, R.L. Graham and D.F. Hsu (eds). Graph Theory, Combinatorics, Algorithms and Applications. Philadelphia: S.I.A.M. Publications, 1991.
    2. K.T. Atanassov. On a Second New Generalization of the Fibonacci Sequence, The Fibonacci Quarterly, 24 (1986): 362-365.
    3. M. Feinberg. Fibonacci-Tribonacci. The Fibonacci Quarterly. 1.3 (1963): 71-74.
    4. A.F. Horadam. A Generalized Fibonacci Sequence. American Mathematical Monthly. 68 (1961): 455-459.
    5. A.F. Horadam. Basic Properties of a Certain Generalized Sequence of Numbers. The Fibonacci Quarterly. 3 (1965): 161-176.
    6. A.G. Shannon. Iterative Formulas Associated with Generalized Third Order Recurrence Relations. S.I.A.M. Journal on Applied Mathematics. 23 (1972): 264-368.
    7. A.G. Shannon and Leon Bernstein. The Jacobi-Perron Algorithm and the Algebra of Recursive Sequences. Bulletin of the Australian Mathematical Society. 8 (1973): 261- 277.
    8. A.G. Shannon and R.S. Melham. Some Aspects of a Partial Difference Equation. Bulletin of Number Theory. 16 (1996): 31-44.
    9. N.J.A. Sloane and Simon Plouffe. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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      Cite this paper

      APA

      Atanassov, K. T., and Shannon, A. G. (2006). Some recurrence relations associated with the Alavi sequence. Notes on Number Theory and Discrete Mathematics, 12(3), 20-24.

      Chicago

      Atanassov, KT, and AG Shannon. “Some Recurrence Relations Associated with the Alavi Sequence.” Notes on Number Theory and Discrete Mathematics 12, no. 3 (2006): 20-24.

      MLA

      Atanassov, KT, and AG Shannon. “Some Recurrence Relations Associated with the Alavi Sequence.” Notes on Number Theory and Discrete Mathematics 12.3 (2006): 20-24. Print.

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