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project-euler / project-euler / 137 / analysis.txt

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F[n] = 1/sqrt(5) * { [ (1 + sqrt(5)) / 2 ]^n - [ (1 - sqrt(5)) / 2 ] ^ n }

$s = sqrt(5)

Therefore:

[ Given limits according to the |q| <= 1 ]

A_F(x) = 1/$s * { x(1+$s)/2/[ 1 - x(1+$s)/2] - x(1-$s)/2/[1 - x(1-$s)/2] } =

x/(2*$s) * { (1+$s)/[1 - x(1+$s)/2] - (1-$s)/[1 - x(1-$s)/2] } =

x/(2*$s) * { 1/[1/(1+$s) - x/2] - 1/[1/(1-$s) - x/2] } =