What is the Correct Formula for the Sequence x_0 in Terms of S?

[JDude13]

I have a pattern which I am having trouble working out the equation for...
It goes:

S=0, x_0=\frac{1}{1}

S=1, x_0=\frac{0}{2}

S=2, x_0=\frac{2}{4}

S=3, x_0=\frac{0}{8}

S=4, x_0=\frac{6}{16}

S=5, x_0=\frac{0}{32}

S=6, x_0=\frac{20}{64}
\vdots

I know it has something to do with
\frac{1+cos(\pi S)}{2}

I have left the fractions unsimplified to show the relationship with the denominator and S as

DENOMINATOR=2^s

 

[Mark44]

Your formula breaks down at $x_4$ and $x_6$. As your formula stands, $x_S = \frac{1 + \cos(\pi S)}{2^S}$, the numerator will always be 0 or 2.