ISL 1979 Problem 45

For any positive integer $n$ , we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$ , we have $F(10) = 4$ . Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$ , but odd if $n$ is divisible by $6$ .