There are similar questions here, but not exactly what I'm looking for. I'm a physics students and we are asked to prove that using the following 6 shapes, The number of ways to build a 2 by N tiling is $e^{cN}$, we are also asked to find c. Also take into consideration that N goes to infinity. Here are the available shapes. I read some material on this topic and I think recursive relations is the way to go, here is my attempt:
Let $A(N)$ be the number of ways to tile a $2\times N$ rectangle. $A(0)=1$, $A(1)=1$. We can use two vertical $1\times 1$ rectangles to build a $2\times 2$ tiling, we can use the two combinations of L shapes to build $2\times 3$ tiling, now for $N\geq4$ we can use the L shapes with the vertical rods to build shapes ($2\times 5$ example in the picture). So the recursive relation I proposed is:
\begin{align*} A(N) = A(N-1)+A(N-2)+2A(N-3)+2A(N-3)+\cdots+2A(0) \end{align*} \begin{align*} A(N) = A(N-1)+A(N-2)+2\sum_{i=0}^{N-3}A(N-3-i) \end{align*}
Plugging in $A(N)=e^{cN}$, taking $N\to \infty$, and doing the infinite geometric series, I arrive at the equation
\begin{align*} e^{3c}-2e^{2c}-1=0 \end{align*}
This equation has one real solution $c\approx0.79$ and two imaginary ones. I think I made a mistake because I shouldn't get imaginary solutions. What seems to be the problem? Thanks!