On fibonacci numbers of the type F_t(k,n)

In this thesis, the research is conducted on distance Fibonacci sequences. In the relevant context, the literature is analyzed and the research is placed into a conceptual framework. It is observed that there is a gap in many aspects of distance Fibonacci sequences and research on this topic is scarce. The focused on closing several gaps of distance Fibonacci sequences in the literature. F_3 (k,n) and F_4 (k,n) distance Fibonacci sequences are defined and their properties are analyzed. Their relationship with the Padovan and Narayana sequences are found for special values of k in these sequences. Various properties of these sequences are given and generator matrices are found for the sequences. Later, the definitions of these sequences are extended to negative numbers and their relationship with Pascal's triangle are found. Following this generalized F_t (k,n) distance Fibonacci sequence are defined and the same steps are followed for these sequences as well.