Altered Numbers of Fibonacci Number Squared

Authors : Fikri Köken, Emre Kankal

Pages : 73-82

Doi:10.53570/jnt.1368751

View : 67 | Download : 57

Publication Date : 2023-12-31

Article Type : Research

Abstract :We investigate two types of altered Fibonacci numbers obtained by adding or subtracting a specific value $\\{a\\}$ from the square of the $n^{th}$ Fibonacci numbers $G^{(2)}_{F(n)}(a)$ and $H^{(2)}_{F(n)}(a)$. These numbers are significant as they are related to the consecutive products of the Fibonacci numbers. As a result, we establish consecutive sum-subtraction relations of altered Fibonacci numbers and their Binet-like formulas. Moreover, we explore greatest common divisor (GCD) sequences of r-successive terms of altered Fibonacci numbers represented by $\\left\\{G^{(2)}_{F(n), r}(a)\\right\\}$ and $\\left\\{H^{(2)}_{F(n), r}(a)\\right\\}$ such that $r\\in\\{1,2,3\\}$ and $a\\in\\{1,4\\}$. The sequences are based on the GCD properties of consecutive terms of the Fibonacci numbers and structured as periodic or Fibonacci sequences.
Keywords : Altered Fibonacci number, greatest common divisor (GCD) sequence, Fibonacci sequence