Number of n x n binary matrices having a contiguous 2 x 2 submatrix whose every element is 1

This sequence is of interest in the theory of 'Reliability in Engineering', where a 2-dimensional m by n array of elements might be considered to have failed if and only if it includes a contiguous r by s rectangle of failed elements. This is an extension of the 1-dimensional problem exemplified by a sequence of pumping stations along a pipeline. In the case where each element fails with probability 1/2, independently of the other elements, computing the probability that the system fails becomes a combinatorial problem.