[PollardsRho]

They then say there's an approximation for Fibonacci, which makes me think that's what they're calling Binet's formula. (I'd also expect an author with this mathematical sophistication to be aware of Binet's formula, but maybe I'm projecting.)

[paulpauper]

Yes, the Binet formula is a closed-form. It arises because of a 2nd order linear recursion , which is not the same as the generating function. Also he confuses the characteristic equation for the generating function. I would recommend a discreate math textbook if you want to learn this.