A New Generalization of Jacobsthal Numbers (bi-periodic Jacobsthal Sequences)

The bi-periodic Fibonacci sequence also known as the generalized Fibonacci sequence was fırst introduced into literature in 2009 by Edson and Yayenie [9] after which the bi-periodic Lucas sequence was defined in a similar fashion in 2004 by Bilgici [5]. In this study, we introduce a new generalization of the Jacobsthal numbers which we shall call bi-periodic Jacobsthal sequences similar to the bi-periodic Fibonacci and Lucas sequences as ̂n = { a̂n−1 + 2̂n−2, if n is even b̂n−1 + 2̂n−2, if n is odd n ≥ 2, with initial conditions ̂0 = 0, ̂1 = 1. we then proceed to find the Binet formula as well as the generating function for this sequence. The well known Cassini, Catalans and the D’ocagne’s identities as well as some related binomial summation formulas were also given. The convergence properties of the consecutive terms of this sequence was also examined.