Properties and Generalizations of Altered Jacobsthal Numbers Squared and their GCD Sequences - Koken, Fikri; Ergun, Halime; Uzun, Yusuf (Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya, 2025)
This paper investigates two types of altered Jacobsthal numbers, namely G(2)J(n) (a) and H(2)J(n) (a), which are obtained by adding or subtracting a specific value, denoted with a, from the square of the nth Jacobsthal numbers. These numbers exhibit a close relationship with the consecutive products of the Jacobsthal numbers. The study establishes consecutive sum-subtraction relations for the altered Jacobsthal numbers, and derives their Binet-like formulas. Furthermore, 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), r ∈ 1, 2, 3, 4 are investigated. It is observed that these sequences display either a periodic or Jacobsthal structure.