Properties and Generalizations of Altered Jacobsthal Numbers Squared and their GCD Sequences - Koken, Fikri; Ergun, Halime; Uzun, Yusuf (Science and Technology Indonesia, 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.