  For any increasing function $f: {\Bbb N} \rightarrow {\Bbb N}_{\ge 2}$ which
takes only finitely many distinct values, a connected finite dimensional
algebra $\Lambda$ is constructed, with the property that $\text{fin.dim}_n\,
\Lambda = f(n)$ for all $n$; here $\text{fin.dim}_n\, \Lambda$ is the
$n$-generated finitistic dimension of $\Lambda$. The stacking technique
developed for this construction of homological examples permits strong control
over the higher syzygies of $\Lambda$-modules in terms of the algebras serving
as layers.
