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).