On combined 3-Fibonacci sequences

Krassimir T. Atanassov, Lilija C. Atanassova and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 758–764
DOI: 10.7546/nntdm.2022.28.4.758-764
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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
Intelligent Systems Laboratory, “Prof. Asen Zlatarov” University
1 “Prof. Yakimov” Blvd., Burgas 8010, Bulgaria

Lilija C. Atanassova
Institute of Information and Communication Technologies,
Bulgarian Academy of Sciences
“Acad. G. Bonchev” Str., Bl. 2, Sofia 1113, Bulgaria

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

Abstract

The term ‘combined’ sequence includes any of the ‘coupled’, ‘intercalated’ and ‘pulsated’ sequences. In this paper, k = 3, so new combined 3-Fibonacci sequences, \{\alpha_n \}, \{ \beta_n \}, \{ \gamma_n \}, are introduced and the explicit formulae for their general terms are developed. That is, there are three such sequences, each with a linear recurrence relation which contains terms from the other two. In effect, each such recurrence relation is second order, with two initial terms which specify the subsequent delineation of the terms of the sequences. The initial terms are, respectively, \langle \alpha_0, \alpha_1 \rangle = \langle 2a, 2d \rangle, \langle \beta_0, \beta_1 \rangle = \langle b,e \rangle and \langle \gamma_0, \gamma_1 \rangle = \langle 2c, 2f \rangle in turn. These result in neat inter-relationships among the three sequences, which can lead to intriguing connections with known sequences, and to a surprisingly simple graphical representation of the whole process. The references include a comprehensive cover of the pertinent literature on these aspects of recursive sequences particularly during the last seventy years.

A secondary goal of the paper is to put the disarray of this part of number theory into some semblance of order with a selection of representative references. This gives rise to a ‘combobulated sequence’, so-called because it restores partial order to a disarray of many papers into three classes, which are fuzzy in both their membership and non-membership because of their diverse and non-systematic derivations.

Keywords

  • Fibonacci sequence
  • 3-Fibonacci sequence
  • Combobulated sequence

2020 Mathematics Subject Classification

  • 11B39

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Manuscript history

  • Received: 20 July 2022
  • Revised: 27 October 2022
  • Accepted: 14 November 2022
  • Online First: 16 November 2022

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Cite this paper

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, DOI: 10.7546/nntdm.2022.28.4.758-764.

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