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

It goes:

[tex]S=0, x_0=\frac{1}{1}[/tex]

[tex]S=1, x_0=\frac{0}{2}[/tex]

[tex]S=2, x_0=\frac{2}{4}[/tex]

[tex]S=3, x_0=\frac{0}{8}[/tex]

[tex]S=4, x_0=\frac{6}{16}[/tex]

[tex]S=5, x_0=\frac{0}{32}[/tex]

[tex]S=6, x_0=\frac{20}{64}[/tex]

[tex]\vdots[/tex]

I know it has something to do with

[tex]\frac{1+cos(\pi S)}{2}[/tex]

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

[tex]DENOMINATOR=2^s[/tex]

It goes:

[tex]S=0, x_0=\frac{1}{1}[/tex]

[tex]S=1, x_0=\frac{0}{2}[/tex]

[tex]S=2, x_0=\frac{2}{4}[/tex]

[tex]S=3, x_0=\frac{0}{8}[/tex]

[tex]S=4, x_0=\frac{6}{16}[/tex]

[tex]S=5, x_0=\frac{0}{32}[/tex]

[tex]S=6, x_0=\frac{20}{64}[/tex]

[tex]\vdots[/tex]

I know it has something to do with

[tex]\frac{1+cos(\pi S)}{2}[/tex]

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

[tex]DENOMINATOR=2^s[/tex]

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