TITLE: Non-trivial examples of Stably diffeomorphic 4-manifolds
QUESTION [8 upvotes]: I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp_n (S^2\times S^2) \cong N \sharp_r (S^2\times S^2)$$ for $r,n$ not necessarily the same

By non trivial I mean that $M$ and $N$ are not diffeomorphic or that $M \cong N\sharp S^ 2 \times S^2$ or similar situation to these two. 
Clearly the two manifolds have to have the same fundamental group and the same signature at least. What I am looking for is a concrete/explicit example.
By results of Gompf we know that if $M$ is orientable carries two non-equivalent smooth structures ($M_1,M_2$) then $M_1$ and $M_2$ are stably diffeomorphic (but clearly non-diffeomorphic). I would like to have a less exotic example as main example in mind when speaking about stable diffeomorphism.
I found abstract criterions for certain families of manifolds but I'm unable to cook up an example.

REPLY [2 votes]: Since the examples given above are smooth, simply connected, and homotopy equivalent, they are also homeomorphic.  You might like the following examples. Let $L$ and $L'$ be 3-dimensional lens spaces that are homotopy equivalent but not homeomorphic. Then $L \times S^1$ and $L' \times S^1$ are simple homotopy equivalent and stably diffeomorphic, but not homeomorphic.