New York Journal of Mathematics Volume 27 (2021), 1115-1133
 Volume 27 Volume Index

Duplications in the k-generalized Fibonacci sequences

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 Published: July 24, 2021. Keywords: k-generalized Fibonacci sequence, multiplicity. Subject: 11B39, 11D61.

Abstract
Let k ≥ 3 be an odd integer. Consider the k-generalized Fibonacci sequence backward. The characteristic polynomial of this sequence has no dominating zero, therefore the application of Baker's method becomes more difficult. In this paper, we investigate the coincidence of the absolute values of two terms. The principal theorem gives a lower bound for the difference of two terms (in absolute value) if the larger subscript of the two terms is large enough. A corollary of this theorem makes possible to bound the coincidences in the sequence. The proof essentially depends on the structure of the zeros of the characteristic polynomial, and on the application of linear forms in the logarithms of algebraic numbers. Then we reduced the theoretical bound in practice for 3 ≤ k ≤ 99, and determined all the coincidences in the corresponding sequences. Finally, we explain certain patterns of pairwise occurrences in each sequence depending on k if k exceeds a suitable entry value associated to the pair.

Acknowledgements

The authors are grateful to A. Mehdaoui, Sz. Tengely, T. Wurth, T. Bartalos, and Gy. Bugar for their kind help in carrying out the computations. F. Luca worked on this paper while he visited Max Planck Institute for Software Systems in Saarbrucken, Germany in the Fall of 2020. He thanks this Institution for hospitality and support. For L. Szalay the research and this work was supported by Hungarian National Foundation for Scientific Research Grant No. 128088, and No. 130909, and by the Slovak Scientific Grant Agency VEGA 1/0776/21.

Author information

Florian Luca:
School of Maths, Wits University
Johannesburg, South Africa;
and Research Group in Algebraic Structures and Applications
King Abdulaziz University, Jeddah, Saudi Arabia;
and Centro de Ciencias Matematicas, UNAM, Morelia, Mexico

Florian.Luca@wits.ac.za

Attila Petho:
University of Debrecen
Department of Computer Science
Debrecen, Hungary

petho.attila@unideb.hu

Laszlo Szalay:
J. Selye University
Department of Mathematics
Komarno, Slovakia;
and University of Sopron
Institute of Mathematics
Sopron, Hungary

szalay.laszlo@uni-sopron.hu