F(n,n)=1^2+{1\over 2}\left (2^2+{1\over 2}\left (3^2+{1\over 2}\left (\ldots +{1\over 2}\left ((n-1)^2 +{n\over 2} \right )\ldots \right )\right )\right )=2S_{n-1}^2+{n\over 2^{n-1}}