In this article, we introduce a new generalization of Fibonacci sequence and we call it as k-Fibonacci-Like sequence. After that we obtain some fundamental properties for k-Fibonacci-Like sequence and also we present some relations among k-Fibonacci-Like sequence, k-Fibonacci sequence and k-Lucas sequence by some algebraic methods.
The Fibonacci sequence is a well-known example of second order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Fibonacci sequence is introduced and dened by H k,n+1 = 2H k,n + kH k,n−1 , n ≥ 1, H k,0 = 2, H k,1 = 1 and k is the positive real number. Also n th power of the generating matrix for this generalized Fibonacci sequence is established and some basic properties of this sequence are obtained by matrix methods.
In the present paper first and foremost we introduce a generalization of a classical Fibonacci sequence which is called a 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.
Abstract. Fibonacci sequence is a well known example of second order linear recurrence relatio. Besides Fibonacci numbers and their generalizations have many applications as well as interesting properties almost in every field of science such as in Physics, Biology, Mathematics (Algebra, Geometry and Number Theory itself). The main aim of the present article to introduce a generalization of Fibonacci sequence which is similar to k-Pell, k-Pell-Lucas, Modified k-Pell sequences and known as Fibonacci-Like sequence. After that we obtain some fundamental properties of Fibonacci-Like sequence such as Binet formulae of Fibonacci-Like sequence, binomial transform of the Fibonacci-Like sequence and sum of Fibonacci-Like numbers with indexes in an arithmetic sequence. In addition to this we obtain some new relations among k-Pell, k-Pell-Lucas, Modified k-Pell and Fibonacci-Like sequences.
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