Two Fibonacci-like sequences

Krassimir T. Atanassov and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 335–339
DOI: 10.7546/nntdm.2025.31.2.335-339
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Authors and affiliations

Krassimir T. Atanassov
Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
Acad. G. Bonchev Str. Bl. 105, Sofia-1113, Bulgaria

Anthony G. Shannon
Warrane College, University of New South Wales
Kensington, NSW 2033, Australia

Abstract

In the present paper we will discuss the two Fibonacci-like sequences

    \[ s_{2k+1} = s_{2k} - s_{2k-1} + \cdots - s_1 + s_0\]

with s_{0} = a_0, \ldots, s_{2k} = a_{2k} and

    \[s_{2k+2} = s_{2k+1} - s_{2k} + \cdots + s_1 - s_0\]

with s_{0} = a_0, \ldots, s_{2k} = a_{2k+1}, where a_0, \ldots, a_{2k+1} are arbitrary numbers.
Examples and explicit formulas for s_{2k+1} and s_{2k+2} are given.

Keywords

  • Arbitrary order recurrence relation
  • Combined sequence
  • Fibonacci sequence
  • Intersection

2020 Mathematics Subject Classification

  • 11B39

References

  1. Atanassov, K. T., Atanassova, L. C., & Shannon, A. G. (2022). On combined 3-Fibonacci sequences. Notes on Number Theory and Discrete Mathematics, 28(4), 758–764.
  2. Atanassov, K. T., Deford, D. R., & Shannon, A. G. (2015). Pulsated Fibonacci recurrences. The Fibonacci Quarterly, 52(5), 22–27.
  3. Atanassova, V. K., Shannon, A. G., & Atanassov, K. T. (2003). Sets of extensions of the Fibonacci sequence. Comptes rendus de l’Académie Bulgare des Sciences, 56(9), 9–12.
  4. Philippou, A. N. (1983). A note on the Fibonacci sequence of order k and the multinomial coefficients. The Fibonacci Quarterly, 21(2), 82–86.
  5. Shannon, A. G. (1976). Some number theoretic properties of arbitrary order recursive sequences. Bulletin of the Australian Mathematical Society, 14(1), 149–151.
  6. Shannon, A. G., Turner, J. C., & Atanassov, K. T. (1991). A generalized tableau associated with coloured convolution trees. Discrete Mathematics, 92, 329–340.
  7. Stein, S. K. (1962). The intersection of Fibonacci sequences. Michigan Mathematical Sequences, 9, 399–402.
  8. Subba Rao, K. (1953). Some properties of Fibonacci numbers – I. Bulletin of the Calcutta Mathematical Society, 46, 253–257.
  9. Williams, H. C. (1972). On a generalization of the Lucas functions. Acta Arithmetica, 20, 33–51.

Manuscript history

  • Received: 30 March 2025
  • Accepted: 17 May 2025
  • Online First: 2 June 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Atanassov, K. T., & Shannon, A. G. (2025). Two Fibonacci-like sequences. Notes on Number Theory and Discrete Mathematics, 31(2), 335-339, DOI: 10.7546/nntdm.2025.31.2.335-339.

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