A Study on Altered Jacobsthal Lucas Numbers Squared: Structural Properties and Applications
Fikri Koken *
Department of Computer Engineering, Faculty of Ahmet Cengiz Engineering, University of Necmettin Erbakan, Konya, Turkey.
*Author to whom correspondence should be addressed.
Abstract
We examine two variations of the Jacobsthal Lucas numbers, denoted as \(G^{(2)}_{j(n)}\)(a) and \(H^{(2)}_{j(n)}\)(a) which are derived through the addition or subtraction of a specific value {a} from the square of the nth Jacobsthal Lucas numbers due to their relevance to the products of Jacobsthal numbers. Consequently we derive both the consecutive sum-subtraction relationships and Binet-like expressions for these altered sequences, while also investigating the greatest common divisor (Gcd) sequences of r–successive terms, represented by {\(G^{(2)}_{j(n)}\),r (a)} and {\(H^{(2)}_{j(n)}\),r (a)} for r ∈ {1, 2, 3, 4}, which are informed by the periodic properties of the Gcd of
consecutive Jacobsthal numbers.
Keywords: Altered jacobsthal lucas number, jacobsthal lucas sequence, jacobsthal sequence, greatest common divisor (Gcd) sequence