How many matrices are possible for the given arrangement? Given m & n, we have to find out the number of possible matrices of order m*n with the property that A(i,j) can be either 0 or 1 and that no contiguous sub-matrix of both length > 1 & breadth > 1 should have same entries i.e. all of its cells shouldn't be 0 or 1. For example if m = 2 & n = 2, the answer is 14: Total possibilities : 2 ^ (2 * 2); Invalid cases: when all 4 cells are 0 or 1. Therefore answer is 2 ^ (2 * 2) - 2 = 14. A sub-matrix of length > 1 & breadth = 1, also breadth > 1 & length = 1 is valid.
 A: Let $a_n$ be the number of $2 \times n$ -matrices avoiding constant 2*2-submatrices.
Then 
$$a_n = \frac{2^{-n} \left(4 \left(17+4 \sqrt{17}\right)
   \left(3+\sqrt{17}\right)^n+\left(\sqrt{17}-17\right)
   \left(\sqrt{17}-3\right)^n e^{i \pi  n}\right)}{17
   \left(3+\sqrt{17}\right)}$$
This should be fairly straightforward to prove,
let $v(n)=(e_{01}(n),e_{10}(n),e_{00}(n),e_{11}(n))$ be the vector of number of $2\times n$-matrices ending with column 01, 10, 00 resp. 11.
We then have the recursion
$$v(n+1)=\begin{pmatrix}  1 & 1 & 1 & 1 \\  1 & 1 & 1 & 1 \\  1 & 1 & 0 & 1 \\  1 & 1 & 1 & 0 \\ \end{pmatrix} v(n)$$
Since this is symmetric, we may diagonalize this and from here, it should be straightforward to find the formula above.
(I cheated a bit in Mathematica).
EDIT: Of course, $e_{01}(n)=e_{10}(n)$ and $e_{00}(n)=e_{11}(n)$ by symmetry,
so one can of course reduce the above to a 2 by 2 matrix recursion instead, with entries 2,2 and 2,1. Eigenvalues of this matrix are $1/2 (3 + \sqrt{17}), 1/2 (3 - \sqrt{17})$
which explains the strange formula above.
A: EDIT :
I've edited the argument to make it stronger
Suppose that $m\geq 3$ and $n\geq 5$ so that there is a 3x5 submatrix A. I show that the number of possibilities is zero in this case.
In A, there are at least two rows with at least there $1$'s each (up to relabeling the symbols). Since we cannot have a constant 2x2 submatrix, we may assume that the first two rows of the matrix are 
[11100] and
[00111].
To avoid a 2x2 contant submatrix, the first two entries of the third row must be different, but then, whatever choice we make for the third one, we will get a constant 2x2-submatrix in the first and third row.
The answer for $m=1$ and $m=2$ is not hard to calculate explicitly.
Together with the answer above, this reduces the problem to checking the following cases (which is not too hard):
3x3,3x4,4x4.
A: Perhaps this could help.
