with initials conditions J(0)((3)) = a, J(1)((3)) = b and J(2)((3)) = c, where a, b and c are non- zero real numbers. Many sequences in the literature are special cases of this sequence. We find the generating function and Binet's formula of the sequence. Then we show that the Cassini and Gelin- Ces` aro identities are satisfied by the indices of this generalized sequence.