Analytic Combinatorics – A Worked Example

Very cool!

What's meant by "it’s already too much to ask for a closed form for fibonacci numbers"? Binet's formula is usually called a closed form in my experience. Is "closed form" here supposed to mean "closed form we can evaluate without needing arbitrary-precision arithmetic"?

Author probably just unaware of Binet's formula. But I'd agree with their assessment that there's unlikely to be a closed form for the (indeed, much more complicated!) quantity that they're interested in.

In fact, for that 'warmup' problem, the leading term has a base and coefficient that are roots of cubic polynomials, given in the OEIS entry. But often the coefficient will be trancendental for these sorts of problems.