f(f(n-1))=f(n+1)-f(n)

if a solution exists (which i somehow doubt but am unable to prove)

At least one solution exists f(n)=0. It is rather trivial, though.

A simple proof by contradiction should at least show that there are no non-trivial linear solutions.

 

(*) f(f(n-1))=f(n+1)-f(n)

assume f to be linear

f²(n-1)=f(1)

f(n-1)=1

f=1

(*) 1=1-1 //

so f is not linear

whoops i forgot to mention the function is from N to N. so the equation is satisfied for n>=2

 

from this i infer f is strict increasing and f(n+1)-f(n)>=n-3 and many more inequalities.

 

but none leads to any sort of contradiction.