Školjka

%V0 Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $$f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1$$ for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$. Proposed by Bojan Bašić, Serbia