On the generalized Fibonacci like sequences and matrices

Abstract

. In this paper, we study the generalized Fibonacci like sequences { 𝑡 𝑘,𝑛 } 𝑘 ∈{ 2 , 3 } ,𝑛 ∈ N with arbitrary initial seed and give some new and well- known identities like Binet’s formula, trace sequence, Catalan’s identity, generating function, etc. Further, we study various properties of these generalized sequences, establish a recursive matrix and relationships with Fibonacci and Lucas numbers and sequence of Fibonacci traces. In this study, we exam-ine the nature of identities and recursive matrices for arbitrary initial values.