  In [25], T. Oikhberg introduced and studied variants of the greedy and weak
greedy algorithms for sequences with gaps, with a focus on the $\mathbf
n$-$t$-quasi-greedy property that is based on them. Building upon this
foundation, our current work aims to further investigate these algorithms and
bases while introducing new ideas for two primary purposes. Firstly, we aim to
prove that for $\mathbf n$ with bounded quotient gaps, $\mathbf
n$-$t$-quasi-greedy bases are quasi-greedy bases. This generalization extends
the result previously established in [7] to the context of Markushevich bases
and, also, completes the answer to a question from [25]. The second objective
is to extend certain approximation properties of the greedy algorithm to the
context of sequences with gaps and study if there is a relationship between
this new extension and the usual convergence.
