TY  - JOUR
T1  - Altered Numbers of Fibonacci Number Squared
AU  - Köken, Fikri
AU  - Kankal, Emre
PY  - 2023
DA  - December
DO  - 10.53570/jnt.1368751
JF  - Journal of New Theory
JO  - JNT
PB  - Naim ÇAĞMAN
WT  - DergiPark
SN  - 2149-1402
SP  - 73
EP  - 82
IS  - 45
LA  - en
AB  - 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.
KW  - Altered Fibonacci number
KW  - greatest common divisor (GCD) sequence
KW  - Fibonacci sequence
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UR  - https://doi.org/10.53570/jnt.1368751
L1  - https://dergipark.org.tr/en/download/article-file/3443917
ER  -