Non-trivial examples of Stably diffeomorphic 4-manifolds 
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.
 A: 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.
A: The main examples of this go back to work of Moishezon and Mandelbaum in the late 1970s. For instance, if a simply-connected elliptic surface E(2n+1) is homotopy equivalent to the connected sum of $4n+1$ copies of $\mathbb{C}P^2$ blown up $20n +9$ times, but those manifolds are not diffeomorphic for $n>1$. This is a theorem of Donaldson. But after a single connected sum with $S^2 \times S^2$ they become diffeomorphic, as proved by Moishezon (with a simplified proof by Mandelbaum-Moishezon).
These matters are described pretty well in the book of Gompf and Stipsicz. As far as I know, there is no known example of a simply connected pair of this sort where you have to add more than one copy of $S^2 \times S^2$.
