Genelleştirilmiş fibonacci p-sayilari ve kombinatoryal yaklaşim ile uygulamalari

In this study, Fibonacci (𝑘, 𝑝)-numbers which generalize both Fibonacci, Jacobsthal, Narayana numbers and Fibonacci 𝑝-numbers in the distance sense are defined by using the definition of the distance between numbers by a recurrence relation according to a new parameter 𝑘. Sequences and combinatorial interpretations of these numbers are presented. Explicit formulas that allow us to calculate the 𝑛th term and generating functions of these sequences are obtained. Also, a tiling approach to the Fibonacci 𝑝-numbers and the Fibonacci (𝑘, 𝑝)-numbers is given. The Fibonacci 𝑝-numbers are interpreted as the number of ways to tile a 1 × 𝑛 board by using various 1 × 𝑟, 𝑟-ominoes from 𝑟 = 1 up to 𝑟 = 𝑝 + 1, and the Fibonacci (𝑘, 𝑝)-numbers are interpreted as the number of ways to tile a 1 × 𝑛 board by using 1 × 1 squares and colored 1 × 𝑝, 𝑝-ominoes, where there are 𝑘 different colors for 𝑝-ominoes. Moreover, product identities and sum formulas of these numbers with special subscripts are given by tiling interpretations that allow the derivation of their properties. 2023, 101 Pages Keywords: Combinatorial formula, Fibonacci number, Generalized Fibonacci numbers, Tilings