On the Adjacency-Jacobsthal numbers

In this article, we define the adjacency-Jacobsthal sequence and then we obtain the combinatorial representations and the sums of adjacency-Jacobsthal numbers by the aid of generating function and generating matrix of the adjacency-Jacobsthal sequence. Also, we derive the determinantal and the permanental representations of adjacency-Jacobsthal numbers by using certain matrices which are obtained from generating matrix of adjacency-Jacobsthal numbers. Furthermore, using the roots of characteristic polynomial of the adjacency-Jacobsthal sequence, we produce the Binet formula for adjacency-Jacobsthal numbers. Finally, we give the relationships between adjacency-Jacobsthal numbers and Fibonacci, Pell, and Jacobsthal numbers.