New families of Jacobsthal and Jacobsthal-Lucas numbers (original) (raw)

On some identities of k-Jacobsthal-Lucas numbers

2014

In this paper we present the sequence of the k-Jacobsthal-Lucas numbers that generalizes the Jacobsthal-Lucas sequence introduced by Horadam in 1988. For this new sequence we establish an explicit formula for the term of order n, the well-known Binet's formula, Catalan's and d'Ocagne's Identities and a generating function.

A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence)

Journal of Advances in Mathematics and Computer Science

In this study, we bring into light a new generalization of the Jacobsthal Lucas numbers, which shall also be called the bi-periodic Jacobsthal Lucas sequence as with initial conditions \ \hat{c}_{0}=2,\ \hat{c}_{1}=a.$$ The Binet formula as well as the generating function for this sequence are given. The convergence property of the consecutive terms of this sequence is examined after which the well known Cassini, Catalan and the D'ocagne identities as well as some related summation formulas are also given.

A New Generalization of Jacobsthal Numbers (Bi-Periodic Jacobsthal Sequences)

2016

The bi-periodic Fibonacci sequence also known as the generalized Fibonacci sequence was fırst introduced into literature in 2009 by Edson and Yayenie [9] after which the bi-periodic Lucas sequence was defined in a similar fashion in 2004 by Bilgici [5]. In this study, we introduce a new generalization of the Jacobsthal numbers which we shall call bi-periodic Jacobsthal sequences similar to the bi-periodic Fibonacci and Lucas sequences as ̂n = { a̂n−1 + 2̂n−2, if n is even b̂n−1 + 2̂n−2, if n is odd n ≥ 2, with initial conditions ̂0 = 0, ̂1 = 1. we then proceed to find the Binet formula as well as the generating function for this sequence. The well known Cassini, Catalans and the D’ocagne’s identities as well as some related binomial summation formulas were also given. The convergence properties of the consecutive terms of this sequence was also examined.

On the Jacobsthal-Lucas Numbers

2008

In this study, we define the Jacobsthal Lucas E-matrix and R-matrix alike to the Fibonacci Q-matrix. Using this matrix represantation we have found some equalities and Binet-like formula for the Jacobsthal and Jacobsthal-Lucas numbers. Mathematis Subject Classification: 11B39; 11K31; 15A24; 40C05

On Jacobsthal–Narayana and Jacobsthal-Narayana-Lucas Sequences

Konuralp Journal of Mathematics, 2024

This paper introduces two new integer sequences that are the third-order recurrence relations. These are called Jacobsthal-Narayana and Jacobsthal-Narayana-Lucas sequences. In particular, great attention is focused on the identification of the Binet type representations for our new sequence, including the generating functions, some important identities, and generating matrix. Finally, we consider the circulant matrix whose entries are Jacobsthal-Narayana sequence and present an appropriate formula to find eigenvalues of that matrix.

The (s, t)-Jacobsthal and (s, t)-Jacobsthal lucas sequences

In this study, two sequences called (s, t)-Jacobsthal, (s, t)-Jacobsthal Lucas are defined by considering the usual Jacobsthal and Jacobsthal Lucas numbers. After that, we establish some properties of these sequences and some important relationships between (s, t)-Jacobsthal sequence and (s, t)-Jacobsthal Lucas sequence.

On the recurrences of the Jacobsthal sequence

Mathematica, 2023

In the present work, two new recurrences of the Jacobsthal sequence are defined. Some identities of these sequences which we call the Jacobsthal array is examined. Also, the generating and series functions of the Jacobsthal array are obtained. MSC 2020. 11B39, 05A15, 11B83.

2 The Jacobsthal and Jacobsthal-Lucas sequences

2012

In the present paper, we define two directed pseudo graph. Then, we investigate the adjacency matrices of the defined graphs and show that the permanents of the adjacency matrices are Jacobsthal and Jacobsthal-Lucas numbers. We also give complex factorization formulas for the Jacobsthal sequence.

On Hyperbolic Jacobsthal-Lucas Sequence

Fundamental Journal of Mathematics and Applications

In this study, we define the hyperbolic Jacobsthal-Lucas numbers and we obtain recurrence relations, Binet’s formula, generating function and the summation formulas for these numbers.

On generalizations of the Jacobsthal sequence

Notes on Number Theory and Discrete Mathematics, 2018

In this paper, the generalized Jacobsthal and generalized complex Jacobsthal and generalized dual Jacobsthal sequences using the Jacobsthal numbers are investigated. Also, special cases of these sequences are investigated. Furthermore, recurrence relations, vectors, the golden ratio and Binet's formula for the generalized Jacobsthal sequences and generalized dual Jacobsthal sequences are given.