Find out how many ways to paint a 1 × n chessboard with an even number of black and any number of white squares

Question

Find out how many ways to paint a $1 × n$ chessboard with an even number of black and any number of white squares using the:

1. exponential stretching function,
2. using a combinatorial argument.

I tried to solve the first one this way: $$B(black)=(1,0,1,0,...)$$ $$W(white)=(1,1,1,1,...)$$ $$B(x)\cdot W(x)=((e^x+e^{-x})\cdot e^x)/2=(e^{2x}+1)/2$$

Here I don't know how to continue .