Altered sequences of second powers of the n.Fibonacci and Lucas numbers and r-consecutive greatest common divisoder sequences

This study consists of five chapters. In the subsections in the first chapter, the divisibility and greatest common divisor properties and the known properties in the literature about Fibonacci and Lucas sequences that will be used in the main results are mentioned. In the second part, similar literature studies about altered Fibonacci and Lucas sequences are mentioned. In the third and fourth sections, which are our main sections, it is seen that sub-Fibonacci and Lucas sequences are formed from altered numbers obtained by adding or subtracting the values   1, 9 from the square of the n. Fibonacci and Lucas numbers, taking into account the properties of Fibonacci and Lucas numbers. The relations of altered sequences with Fibonacci and Lucas sequences are obtained. Therefore, it is shown that these altered sequences are at least as important as the Fibonacci and Lucas numbers. By making use of the reduction and divisibility properties of the Fibonacci and Lucas sequences; Similar reduction relations of altered sequences and Binet formulas are given. Similarly, Fibonacci and Lucas subsequence connections are investigated by searching the largest common divisor sequences of r–consecutive terms on altered numbers according to the greatest common divisor properties of consecutive terms of Fibonacci and Lucas numbers. In the last section, the results and suggestions obtained according to the research, findings and observations made during the study are given.