A Study on Altered Jacobsthal Lucas Numbers Squared: Structural Properties and Applications
Résumé
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 ofconsecutive Jacobsthal numbers.