MathDB

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

R_{mn}

8