On Generalized One-kZero Numbers

In this study is introduced a novel generalization of repunit and one-zero numbers through the formulation of the generalized One-kZero. This sequence extends the classical families of repunit and one-zero numbers by establishing a unified framework in which the parameter (k≥0) specifies the number of consecutive zeros separating two ones in the decimal representation. We introduce the new family of sequences, the generalized One-kZero numbers, and investigate some of their properties. The main purpose is to present a generalization for the recurrence relation of kind One-Zero numbers and determine some relations and properties. The reason that led us to this method is that the recurrence relation of One-Zero and Repunit numbers has a second-order difference equation as a specific case of the Horadam-type sequence. The Binet formula, generating function, sum formulas and many other relations will therefore be much easier to find. Also, some other identities that have not been found before in the particular case of One-Zero and Repunit sequences are also included in this study.