On the generating function of the Fibonacci numbers

Let’s define the Fibonacci numbers as $F_0=1$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. Using this recurrence I was able to calculate the generating function of the Fibonacci numbers to be $-\frac{1}{x^2+x-1}$. Now, it can be proved that $F_n$ counts the list of $1,2$ with sum $n$. Is there a way to find the generating function using this model without using the recurrence?

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