  We extend our investigation of $2$-determinants, which we defined in a
previous paper. For a linear homogenous recurrence of the second order, we
consider relations between different sequences satisfying the same linear
homogeneous recurrence of the second order. After we prove a generalized
identity of d'Ocagne, we derive, from a single identity, a number of classical
identities (and their generalizations) such as d'Ocagne's, Cassini's, Catalan's
and Vajda's. Along the way, the corresponding combinatorial interpretations in
terms of restricted words over a finite alphabet are stated for the sequences
we investigate.
