The adjacency-Jacobsthal-Hurwitz sequences in groups

On fourth-order jacobsthal quaternions

“…The n-th fourth-order Jacobsthal quaternion JQ (4) n and the n-th fourth-order Jacobsthal-Lucas quaternion jQ (4) n can be defined as…”

“…n and j (4) n are the n-th fourth-order Jacobsthal and n-th fourth-order Jacobsthal-Lucas numbers, respectively. Furthermore, the basis i, j, k satisface the rules in (1.8).…”

“…Furthermore, the basis i, j, k satisface the rules in (1.8). The function G(t) = ∑ n≥0 JQ (4) n t n is called the generating function for the sequence {JQ (4) n }. In [3], the authors found a generating function for fourth-order Jacobsthal numbers.…”

“…The Jacobsthal numbers have many interesting properties and applications in many fields of science (see, e.g. [1,4,12,9]). In [1], Barry investigated a Jacobsthal decomposition of Pascal's triangle.…”

“…, r − 1. In this work we consider the particular case r = 4, the fourth-order Jacobsthal numbers {J (4) n } n≥0 and the fourth-order Jacobsthal-Lucas numbers { j (4) n } n≥0 are defined by…”