Query: Jacobsthal sequences

title: On the Quadra Fibona Pell and Hexa Fibona Pell Jacobsthal Sequences
abstract: In this paper, we consider Fibonacci, Lucas, Pell, Pell Lucas, Jacobsthal, Jacobsthal Lucas sequences. We introduce the quadra Fibona Pell,Fibona Jacobsthal and Pell Jacobsthal and the hexa Fibona Pell Jacobsthal sequences whose components are the Fibonacci, Pell and Jacobsthal sequences. We derive the Binet like formulas, the generating functions and the exponential generating functions of these sequences. Also, we obtain some binomial identities for them.

title: The Gelin Cesaro identity in some third order Jacobsthal sequences
abstract: In this paper, we deal with two families of third order Jacobsthal sequences. The first family consists of generalizations of the Jacobsthal sequence. We show that the Gelin Ces\`aro identity is satisfied. Also, we define a family of generalized third order Jacobsthal sequences \mathbb{J}_{n}(3)}_{n\geq 0} by the recurrence relation \mathbb{J}_{n+3}(3)\mathbb{J}_{n+2}(3)\mathbb{J}_{n+1}(3)+2\mathbb{J}_{n}(3) n\geq0, with initials conditions \mathbb{J}_{0}(3)=a$ \mathbb{J}_{1}(3)=b$ and \mathbb{J}_{2}(3)=c$ where $a$ $b$ and $c$ are non zero real numbers. Many sequences in the literature are special cases of this sequence. We find the generating function and Binet's formula of the sequence. Then we show that the Cassini and Gelin Ces\`aro identities are satisfied by the indices of this generalized sequence.

title: A NEW GENERALIZATION OF JACOBSTHAL NUMBERS (BI PERIODIC JACOBSTHAL SEQUENCES)
abstract: The bi periodic Fibonacci sequence also known as the generalized Fibonacci sequence was first introduced into literature in 2009 by Edson and Yayenie  after which the bi periodic Lucas sequence was defined in a similar fashion in 2004 by Bilgici  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 an 1 2n 2, if n is even bn 1 2n 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.

title: The Jacobsthal Sequences in Finite Groups
abstract: In this paper, we study the generalized order Jacobsthal sequences modulo for and the generalized order k Jacobsthal Padovan sequence modulo for Also, we define the generalized order k Jacobsthal orbit of a k generator group for and the generalized order k Jacobsthal Padovan orbit a k generator group for Furthermore, we obtain the lengths of the periods of the generalized order 3 Jacobsthal orbit and the generalized order 3 Jacobsthal Padovan orbit of the direct product, and the semidirect product

title: Binominal transforms of k Jacobsthal sequences
abstract: In this paper, we define the binomial, k binomial, rising, and falling transforms for k Jacobsthal sequence. We investigate some properties of these sequences such as recurrence relations, Binet's formula, generating functions and in the sequel of this paper denote Pascal Jacobsthal triangle for all binomial transformation sequences.

title: Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes
abstract: This study involves definitions of regular and representational multiple counting Jacobsthal sequences of Carmichael numbers. We introduce recurrence relations for multiple counting Jacobsthal sequences and show their association with Fermat's little theorem. We also provide matrix representations and generalized Binet formulas for defined sequences. This leads to a better understanding of how certain composite numbers are distributed among consecutive powers.

title: The Jacobsthal Sequences in The Groups Q2n, Q2nXqZ2m and Q2nXZ2m
abstract: In  Deveci et.al defined the generalized order k Jacobsthal orbit kG of a finitely generated group GA J A, where ka ,ka A1,,a a2, to be the sequence of the elements of G such that ix

title: The Relation Between Chebyshev Polynomials and Jacobsthal and Jacobsthal Lucas Sequences
abstract: In this paper Jacobsthal, Jacobsthal Lucas and generalized Jacobsthal sequences are denoted by aid of first or second type of Chebyshev polynomials by different equalities. Then using these equalities a relation is obtained between Jacobsthal and generalized Jacobsthal numbers. Moreever, the nth powers of some special matrices are found by using Jacobsthal numbers or Chebyshev polynomials. Some connections among Jacobsthal, Jacobsthal Lucas are revealed by using the determinant of the power of some special matrices. Then, the properties of Jacobsthal, Jacobsthal Lucas numbers are obtained by using the identities of Chebyshev polynomials.

title: Bivariate Jacobsthal and Jacobsthal Lucas polynomial sequences
abstract: In this study, we define and study the generalized bivariate Jacobsthal and Jacobsthal Lucas polynomial sequences. The Binet formulae, generating functions, some sum formulae, and interesting properties of these sequences are given.

title: On Third Order \overline{h} Jacobsthal and Third Order \overline{h} Jacobsthal Lucas Sequences, and Related Quaternions
abstract: In this paper, inspired by recent articles of G. Anatriello and G. Vincenzi (see \cite{Ana} and G. Cerda Morales (see \cite{Cer1,Cer2} we will introduce the third order \overline{h} Jacobsthal and third order \overline{h} Jacobsthal Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in \cite{Cer2}