$f(x) * f(y) = f(x-y) + f(x+y)$

Question

It's a question from a Math Olympiad in Azerbaijan. There is a function $f :Z \rightarrow Z$ and $f(x) * f(y) = f(x-y) + f(x+y)$. If $f(1) = 1$, find $f(100)$

Answer 1

My answer was that the there is no sequence $a_1,a_2,\cdots$ for what $a_n = a_{n-1} + a_{n+1}$.

Answer 2

How I would begin (assuming you meant $f(1)=1$): $x=1,y=0$ gives $$ f(1)\cdot f(0)=f(1)+f(1)\\ f(0)=2 $$ Now $x=1,y=1$ gives $$ f(1)\cdot f(1)=f(2)+f(0)\\ f(2)=-1 $$ From here, can you find $f(3)$? What about $f(4)$? Can you guess a pattern, and prove that it holds? Or do you have to calculate a hundred different values? Or can you, at least, be clever about which values of $f$ you calculate and get away with much fewer than a hundred calculations to reach $f(100)$?