Bounding the stable genera of Heegaard splittings from below

We describe for each postive integer $k$ a 3-manifold with Heegaard surfaces of genus $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We also show that for every positive $n$, there is a 3-manifold that has $n$ pairwise non-isotopic Heegaard splittings of the same genus all of which are stabilized.