<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value <italic>x</italic> and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of <italic>a</italic>-Fibonacci and <italic>a</italic>-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.</p></abstract>

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

On some new results for the generalised Lucas sequences

Faculty of Mathematics and Computer Science

Department of Electronics, Computing and Mathematics

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On some new results for the generalised Lucas sequences"}

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under...

On some new results for the generalised Lucas sequences

Published Online: Apr 13, 2021

Page range: 17 - 36

Received: Apr 01, 2020

Accepted: May 25, 2020

DOI: https://doi.org/10.2478/auom-2021-0002

Keywords
Generalised Lucas sequence, Generalised Pell-Lucas sequence, Pell equation, Special Pell Equation, Negative Pell equation

© 2021 Dorin Andrica et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On some new results for the generalised Lucas sequences

Published Online: Apr 13, 2021

Page range: 17 - 36

Received: Apr 01, 2020

Accepted: May 25, 2020

DOI: https://doi.org/10.2478/auom-2021-0002

KeywordsGeneralised Lucas sequence, Generalised Pell-Lucas sequence, Pell equation, Special Pell Equation, Negative Pell equation

© 2021 Dorin Andrica et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Generalised Lucas sequence, Generalised Pell-Lucas sequence, Pell equation, Special Pell Equation, Negative Pell equation