  Given a graded ring $A$ and a homogeneous ideal $I$, the ideal is said to be
of linear type if the Rees algebra of $I$ is isomorphic to the symmetric
algebra of $I$. We examine the relations between different notions of sequences
already known in literature which form ideals of linear type. We observe how
the vanishing of $y$-regularity of the Rees algebra of an ideal is connected to
the ideal being generated by some specific sequences. As an application of
this, we discuss Koszulness and Cohen-Macaulayness of the diagonal subalgebras
of Rees algebras of ideals generated by some of these sequences. We also
mention Macaulay2 algorithms to check if an ideal is generated by a
$d$-sequence, c-sequence, weak relative-regular sequence and for computing
bi-regularities.
