Let U = x + y,       V = x*y    , S(n) = x^n + y^n

S(n)	= U*S(n-1) - V*S(n-2)
S(n-1)= U*S(n-2) - V*S(n-3)
.
.
S(6)	= U^6 - 6*V*U^4 + 9*V^2*U^2 - 2*V^3
S(5)	= U^5 - 5*V*U^3 + 5*V^2*U
S(4)	= U^4 - 4*V*U^2 + 2*V^2
S(3)	= U^3 - 3*U*V		= U^3 - 3*U*V
S(2)	= x^2 + y^2			= U^2 - 2*V
S(1)	= x + y			= U
S(0)	= 2  

For odd n

F(n)	= x^(n-1) - y*x^(n-2) + y^2*x^(n-3) .... -x*y^(n-2) + y^(n-1)         
	= S(n-1) - V*S(n-3) + V^2*S(n-5) .....+ V^((n-3)/2)*S(1) - V^((n-1)/2)
 	= S(n-1) - V*S(n-3) + V^2*S(n-5) .....+ V^((n-3)/2)*U - V^((n-1)/2)
	= U*S(n-2) - 2*V*S(n-3) + V^2*S(n-5) + V^((n-1)/2)*U - V^((n-1)/2)

F(n)	= S(n-1) - V*F(n-2)

F(7)	= S(6) - V*F(5)		= U^6 - 7*V*U^4 + 14*V^2*U^2 - 7*V^3  
F(5)	= S(4) - V*F(3) 		= U^4 - 5*V*U^2 + 5*V^2 
F(3)	= U^2 - 3*V
F(1)	= 1



F(n)	= U*(linear function in U and V) plus or - n*V^((n-1)/2)
	= (not relatively prime to U) - (relatively prime to U)
	= (relatively prime to U)

Which is what was required.