On the complex k-Fibonacci numbers

Abstract

We first study the relationship between the k-Fibonacci numbers and the elements of a subset of Q(2). Later, and since generally studies that are made on the Fibonacci sequences consider that these numbers are integers, in this article, we study the possibility that the index of the k-Fibonacci number is fractional; concretely, 2n+1/2. In this way, the k-Fibonacci numbers that we obtain are complex. And in our desire to find integer sequences, we consider the sequences obtained from the moduli of these numbers. In this process, we obtain several integer sequences, some of which are indexed in The Online Enciplopedy of Integer Sequences (OEIS).