(why) is this ratio the golden ratio?

Question

Looking at a slight variation of the fibonacci sequence
f(x) = f(x-1) + f(x-2) + 1
where f(1) = 1, f(2) = 1

I'm trying to find the ratio of this sequence but can't figure out how. To get an approximation I just tried looking at some random examples, (i.e. f(10)/f(9), f(20)/f(19))
And it seems that this is the golden ratio, but I can't understand how that is correct as this sequence seems to grow much faster than the fibonacci sequence

Answer 1

If you write $g(x)= f(x)+1$ you got $$g(x+1)=g(x)+g(x-1)$$