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).

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.