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The Square Terms in Lucas Sequences

Journal of Number Theory, 1996
Let \(P\) and \(Q\) be relatively prime odd integers and define the sequences \(\{U_n\}\) and \(\{V_n\}\) by \(U_n = PU_{n - 1} - QU_{n - 2}\) with \(U_0 = 0\), \(U_1 = 1\) and \(V_n = PV_{n - 1} - QV_{n - 2}\) with \(V_0 = 2\), \(V_1 = P\). The main results of the paper are the following. (i) If \(V_n\) is a square, then \(n = 1,3\) or 5.
Ribenboim, Paulo, McDaniel, Wayne L.
openaire   +2 more sources

Repdigits in k-Lucas sequences

Proceedings - Mathematical Sciences, 2014
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Bravo, Jhon J., Luca, Florian
openaire   +1 more source

Copper ratio obtained by generalizing the Fibonacci sequence

AIP Advances
In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and ...
Engin Özkan, Hakan Akkuş
doaj   +1 more source

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

Journal of New Theory, 2023
In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions.
Paula Maria Machado Cruz Catarino, Orhan Dışkaya, Hamza Menken   +2 more
doaj   +1 more source

On a unified approach to homogeneous second-order linear difference equations with constant coefficients and some applications

Special Matrices
In this article, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory.
Kaddoura Issam, Mourad Bassam
doaj   +1 more source

On the emergence of the Quanta Prime sequence

Arab Journal of Basic and Applied Sciences
This paper presents the Quanta Prime Sequence (QPS) and its foundational theorem, showcasing a unique class of polynomials with substantial implications.
Moustafa Ibrahim
doaj   +1 more source

The relations between bi-periodic jacobsthal and bi-periodic jacobsthal lucas sequence

Cumhuriyet Science Journal, 2021
In this paper, one of the special integer sequences, Jacobsthal and Jacobsthal Lucas sequences which are encountered in computer science is generalized according to parity of the index of the entries of the sequences, called bi-periodic Jacobsthal and ...
Şükran Uygun
doaj   +1 more source

Quaternion-Type Catalan Transforms of the ρ-Fibonacci and ρ-Lucas Numbers

Journal of Mathematics, 2023
In this paper, we define a new sequence called the quaternion-type Catalan sequence and give generating function, exponential representation, quaternionic Catalan matrix and its some properties.
Kübra Gül
doaj   +1 more source

On fourth-order jacobsthal quaternions

Journal of Mathematical Sciences and Modelling, 2018
In this paper, we present for the first time a sequence of quaternions of order 4 that we will call the fourth-order Jacobsthal and the fourth-order Jacobsthal-Lucas quaternions. In particular, we are interested in the generating function, Binet formula,
Gamaliel Cerda-morales
doaj   +1 more source

THE SQUARE TERMS IN GENERALIZED LUCAS SEQUENCES

Mathematika, 2013
Summary: Let \(P\) and \(Q\) be non-zero integers. The generalized Fibonacci sequence \(\{U_n\} \) and Lucas sequence \(\{V_n\}\) are defined by \(U_0 = 0\), \(U_1 = 1\) and \(U_{n+1} = PU_n + QU_{n-1}\) for \(n\geq 1\) and \(V_0 = 2\), \(V_1 = P\) and \(V_{n+1} = PV_n + QV_{n-1} \) for \(n\geq 1\), respectively.
Şiar, Zafer, Keskin, Refik
openaire   +4 more sources