On the properties of Lucas numbers with binomial coefficients (original) (raw)

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.

On the sequences related to Fibonacci and Lucas numbers

J. Korean Math. Soc, 2005

In this paper, we obtain some properties of the sequences U q n and V q n introduced in [6]. We find polynomial representations and formulas of them. For q = 5, U 5 n is the Fibonacci sequence Fn and V 5 n is the Lucas sequence Ln.

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.

On the properties of k-Fibonacci and k-Lucas numbers

In this paper, some properties of k−Fibonacci and k−Lucas numbers are derived and proved by using matrices S =  k 2 1 2 k 2+4 2 k 2  and M =  0 1 k 2 + 4 0 ‹ . The identities we proved are not encountered in the k−Fibonacci and k−Lucas numbers literature.

Some New Generalizations Of The Lucas Sequence

2017

In this paper, we investigate the generalized Lucas, the generalized complex Lucas and the generalized dual Lucas sequence using the Lucas number. Also, we investigate special cases of these sequences. Furthermore, we give recurrence relations, vectors, the golden ratio and Binet's formula for the generalized Lucas and the generalized dual Lucas sequence.

Generalized Lucas Numbers and Relations with Generalized Fibonacci Numbers

2011

In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas numbers and generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers.

On the Binomial Sums of k-Fibonacci and k -Lucas sequences

2011

The main purpose of this paper is to establish some new properties of k-Fibonacci and k-Lucas numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between k-Fibonacci and k-Lucas numbers are revealed to get a more strong result.