The GCD sequences of the altered Lucas sequences - Koken, Fikri (Annales Mathematicae Silesianae, 2020)
In this study, we give two sequences Ln+n≥1 and Ln-n≥1 derived by altering the Lucas numbers with ±1, ±3, terms of which are called as altered Lucas numbers. We give relations connected with the Fibonacci Fn and Lucas Ln numbers, and construct recurrence relations and Binet’s like formulas of the Ln+ and Ln- numbers. It is seen that the altered Lucas numbers have two distinct factors from the Fibonacci and Lucas sequences. Thus, we work out the greatest common divisor (GCD) of r-consecutive altered Lucas numbers. We obtain r-consecutive GCD sequences according to the altered Lucas numbers, and show that their GCD sequences are unbounded or periodic in terms of values r.