  In this paper, we deal with two families of third-order Jacobsthal sequences.
The first family consists of generalizations of the Jacobsthal sequence. We
show that the Gelin-Ces\`aro identity is satisfied. Also, we define a family of
generalized third-order Jacobsthal sequences $\{\mathbb{J}_{n}^{(3)}\}_{n\geq
0}$ by the recurrence relation
$$\mathbb{J}_{n+3}^{(3)}=\mathbb{J}_{n+2}^{(3)}+\mathbb{J}_{n+1}^{(3)}+2\mathbb{J}_{n}^{(3)},\
n\geq0,$$ with initials conditions $\mathbb{J}_{0}^{(3)}=a$,
$\mathbb{J}_{1}^{(3)}=b$ and $\mathbb{J}_{2}^{(3)}=c$, 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.
