This is related to How many sequences of rational squares are there, all of whose differences are also rational squares?

Are there infinite sequences $a_n$ of rational cubes whose first differences are positive squares?

Fixing $a_n$ leads to finding points on the elliptic curve $y^2=x^3-a_n^3$. If it is of positive rank there will be infinitely many choices for $a_{n+1}$.

Is there an explicit construction that avoids finding points on curves?

What about infinite sequence of *integer* cubes -- in this case fixing $a_n$ leads to finding integral points on elliptic curves and these are finite.

A possible generalization might be:

Are there infinite sequences of numbers of type $X$ (e.g. triangular, Fibonacci) whose first differences are positive squares?

I am mainly interested in explicit constructions like the case for $X=\square$.