AUSTRIAN-POLISH 2005 Problem 8

Given the sets $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$ , consider functions $f:R_{mn}\to \{-1,0,1\}$ with the following property: for each quadruple of points $A_1,A_2,A_3,A_4\in R_{mn}$ which form a square with side length $0<s<3$ , we have $$f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.$$ For each pair $(m,n)$ of positive integers, determine $F(m,n)$ , the number of such functions $f$ on $R_{mn}$ .