Baltic Way 2022 · Problem 7

The writer Arthur has $n\ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-authors, each co-author writes on a note how many books they remember having written with Arthur. Inspecting the numbers on the notes, they discover that the numbers written down are the first $n$ Fibonacci numbers (defined by $F_1=F_2=1$ and $F_{k+2}=F_{k+1}+F_k$). For which $n$ is it possible that none of the co-authors had a lapse of memory?