I read this QA answer and has problems about how it is solved.
The above referenced post's problem is: How many bit strings of length four do not have two consecutive 1s?
Since generating function is done by letting polynomial coefficients be corresponding sequence items. Then how to get the following formula?
Let $x$ represent the atom '$0$' and $x^2$ represent the atom '$10$' and build all possible strings by concatenating one or more atoms and removing the rightmost '$0$'. $$ \begin{align} \overbrace{\vphantom{\frac1x}\ \ \ \left[x^4\right]\ \ \ }^{\substack{\text{strings of}\\\text{length $4$}}}\overbrace{\ \quad\frac1x\ \quad}^{\substack{\text{remove the}\\\text{rightmost '$0$'}}}\sum_{k=1}^\infty\overbrace{\vphantom{\frac1x}\left(x+x^2\right)^k}^\text{$k$ atoms} &=\left[x^4\right]\frac{1+x}{1-x-x^2}\\ &=\left[x^4\right]\left(1+2x+3x^2+5x^3+8x^4+13x^5+\dots\right)\\[9pt] &=8 \end{align} $$ Note that the denominator of $1-x-x^2$ induces the recurrence $a_n=a_{n-1}+a_{n-2}$ on the coefficients.