On Jacobsthal Binary Sequences

S. Magliveras and W. Wei∗, Florida Atlantic University Let Σ = {0, 1} be the binary alphabet, and A = {0, 01, 11} the set of three strings 0, 01, 11 over Σ. Let A∗ denote the Kleene closure of A, and Z the set of positive integers. A sequence in A∗ is called a Jacobsthal binary sequence. The number of Jacobsthal binary sequences of length n ∈ Z is the n Jacobsthal number. Let k ∈ Z, 1 ≤ k ≤ n. The number of Jacobsthal binary sequences with 1 at the k position from the left is denoted by an,k. A formula for this number has been derived recently. In this paper we consider the general case of a(n; k1, k2, . . . , km), the number of Jacobsthal binary sequences with 1 at each of the k i (1 ≤ i ≤ m) positions from the left, where m, ki ∈ Z; 1 ≤ m < n; 1 ≤ k1 < k2 < . . . < km ≤ n. We present a formula for a(n; k1, k2, . . . , km), and study some other special types of Jacobsthal binary sequences. Some identities involving these numbers are also given.