On the k-Mersenne–Lucas numbers

Abstract

In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.

On The Hyperbolic k-Mersenne And k-Mersenne-Lucas Octonions

“…They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. Mersenne numbers have been studied in the literature and various generalizations such as Mersenne-Lucas, k-Mersenne, k-Mersenne-Lucas have been studied [1,4,6,7,17,22,[25][26][27].…”

On $k$-Mersenne and k-Mersenne-Lucas Octonions

“…Motivated essentially by recent works on octonions with the components from a recursive sequence, here we are considering the generalized recursive sequences so-called the k-Mersenne sequence and the k-Mersenne-Lucas sequence, a generalization of the Mersenne sequence. Many papers are dedicated to Mersenne sequence and their generalizations (see, for example [5,11,15,18]). Das ¸demir and Göksal [9] have defined Mersenne quaternions and obtained Binet's formula and generating function of them.…”

On Generalized p-Mersenne Numbers

“…Mersenne sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to this sequence, see for example, [1,2,3,4,5,6,7,8,9,10,15,16,17,18,19,20,22,23,27,28].…”

“…Generalizations of Mersenne numbers can be obtained in various ways (see for example [5,10,17,22]). Our generalizations of Mersenne numbers in Section 2 are not Mersenne numbers in the sense of [10,22] and are Mersenne numbers in the sense of [23] which is given as: a generalized Mersenne sequence {W n } n≥0 = {W n (W 0 , W 1 )} n≥0 is defined by the second-order recurrence relation…”