The GCD Sequences of the Altered Lucas Sequences

In this study, we give two sequences {L⁺_n}_n≥1 and {L⁻_n}_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 F_n and Lucas L_n numbers, and construct recurrence relations and Binet’s like formulas of the L⁺_n and L⁻_n 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.