On the sequences related to Fibonacci and Lucas numbers (original) (raw)

Generalized Fibonacci – Lucas sequence its Properties

Global Journal of Mathematical Analysis, 2014

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 F F n 2 n n-1 n-2 ,    and F 0,F 1 01  , where F n are an n th number of sequences. The Lucas Sequence is defined by the recurrence formula L L L n 2 n n-1 n-2 ,    and L =2, L =1 01 , where L n an nth number of sequences are. In this paper, we present generalized Fibonacci-Lucas sequence that is defined by the recurrence relation 12 B B B n n n   , 2 n  with B 0 = 2s, B 1 = s. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.

Generalizations of Fibonacci and Lucas sequences

Note di Matematica, 2002

In this paper, we consider the Hecke groups H( √ q), q ≥ 5 prime number, and we find an interesting number sequence which is denoted by dn. For q = 5, we get d2n = L2n+1 and d2n+1 = √ 5F2n+2 where L2n+1 is (2n + 1)th Lucas number and F2n+2 is (2n + 2)th Fibonacci number. From this sequence, we obtain two new sequences which are, in a sense, generalizations of Fibonacci and Lucas sequences.

Generalized Fibonacci-Lucas Sequence

Turkish Journal of Analysis and Number Theory, 2014

The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula − − = + , 2 n ≥ with B 0 = 2b, B 1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.

On a Fibonacci-Like Sequence Associated with K-Lucas Sequence

Acta Universitatis Apulensis

In the present article we consider a new generalization of classical Fibonacci sequence and we call it as Fibonacci-Like sequence V k,n and then we study Fibonacci-Like sequence V k,n and k-Lucas sequence L k,n side by side by introducing two special matrices for these two sequences. After that by using these matrices we obtain Binet formulae for Fibonacci-Like sequence and for k-Lucas sequence, we also give Cassini's identity for Fibonacci-Like sequence.

On the Relations between Lucas Sequence and Fibonacci-like Sequence by Matrix Methods

International Journal of Mathematical Sciences and Computing, 2017

In the present paper first and foremost we introduce a generalization of a classical Fibonacci sequence which is known as Fibonacci-Like sequence and at hindmost we obtain some relationships between Lucas sequence and Fibonacci-Like sequence by using two cross two matrix representation to the Fibonacci-Like sequence. The most worth noticing cause of this article is our proof method, since all the identities are proved by using matrix methods.

On generalized Fibonacci and Lucas polynomials

Chaos, Solitons & Fractals, 2009

Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.

A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients

Applied Mathematics and Computation, 2013

In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.