@article{article_1368751, title={Altered Numbers of Fibonacci Number Squared}, journal={Journal of New Theory}, pages={73–82}, year={2023}, DOI={10.53570/jnt.1368751}, author={Köken, Fikri and Kankal, Emre}, keywords={Altered Fibonacci number, greatest common divisor (GCD) sequence, Fibonacci sequence}, 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.}, number={45}, publisher={Naim ÇAĞMAN}