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

F(5)	= S(4) - V*F(3) = U^4 - 4*V*U^2 +2*V^2 - V*(U^2 - 3*V)
	= U^4 - 5*V*U^2 +5*V^2 
F(3)	= U^2 - 3*V


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