On the Third-Order Jacobsthal and Third-Order Jacobsthal–Lucas Sequences and Their Matrix Representations

Abstract

In this paper, we first give new generalizations for third-order Jacobsthal {J (3) n } n∈N and third-order Jacobsthal-Lucas {j (3) n } n∈N sequences for Jacobsthal and Jacobsthal-Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of {J (3) n } n∈N and {j (3) n } n∈N . Then we investigate their properties.

Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences

“…There have been many studies on the generalized Tribonacci numbers V n (x, y, z; a, b, c). For more information, please refer to [3,9,30,31,33,34] and closely related references therein. Some special cases of the generalized Tribonacci sequence V n (x, y, z; a, b, c) are as follows:…”

On Third-Order $\overline{h}$-Jacobsthal and Third-Order $\overline{h}$-Jacobsthal--Lucas Sequences, and Related Quaternions

“…In [6], the authors provided many basic identities for third-order Jacobsthal numbers, {J (3) n } n≥0 , and third-order Jacobsthal-Lucas numbers, {j (3) n } n≥0 (see also [5] and reference therein):…”

On Generalized Third-Order Jacobsthal Numbers

“…It is well-known that the Jacobsthal sequence (sequence A001045 in [5]) {Jn} is defined recursively by the equation, for n ≥ 0 Jn+2 = Jn+1 + 2Jn in which J0 = 0 and J1 = 1. Then Jacobsthal sequence (second order Jacobsthal sequence) is 0, 1, 1, 3,5,11,21,43,85,171,341,683,1365,2731, 5461, 10923, ... This sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example, [6,7,8,9,10,11,12,13,14,15,16,17,18].…”