Generalized Fibonacci – Lucas sequence its Properties

Abstract

Sequences have been fascinating topic for mathematicians for centuries. The Fibonacci sequences are a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci number, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula F

A novel public key cryptography based on generalized Lucas matrices

“…Bob first choose an integer D is such that 1 < D < φ(37) = 36, say D = 10. The set of primitive roots of 37 is given by X= {2, 5, 13,15,17,18,19,20,22,24,32,35 }. Now Bob selects a primitive root α = 17 of p from X.…”

“…E. Özkan, et.al [14] obtained the terms of n-step Lucas polynomials by using matrices & generalizing the concept and then establishing the relationship between Lucas polynomials and Fibonacci polynomials. In [17] Singh, et.al presented a generalization of Fibonacci-Lucas sequences, constructed matrix of order three and established some identities but they followed it for second order sequences, which can be further extended to higher order sequences and matrix construction.…”

Identities of Generalized Fibonacci-Like Sequence

“…T. Koshy [13] explained two chapters on the use of matrices and determinants. Many determinant identities of generalized Fibonacci sequence are discussed in [4], [6] and [11]. In this section some determinant identities of Generalized Fibonacci-Like sequence are presented.…”

On Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas \(\sum_{k=0}^{n}kx^{k}W_{k}\) and \(\sum_{k=1}^{n}kx^{k}W_{-k}\)

“…with the initial values W 0 , W 1 , W 2 , W 3 are arbitrary complex (or real) numbers not all being zero and r, s, t, u are complex numbers. 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 [1][2][3][4][5][6].…”