It is also possible to reduce to 9 Mordell equations. Denoting $x := 2^{[i/3]}3^{[k/3]}$, we get an equation:
$$d x^3 + 1 = y^2,$$
where $d\mid 2^23^2$.
Equivalently,
$$(dy)^2 = (dx)^3 + d^2,$$
which are Mordell equations with known solutions.