Hypothesis: the generator separately tracks the numerator and denominator and uses the xₙ₊₁ = 2*sqrt(1 - xₙ) rule exactly when this will result in both the numerator and denominator remaining integers.
Here is a plot of denominator ( of the closest fraction with denominator < 1000,000,000)
This looks exactly what you would expect if you started with a number that happened to be a fraction, and applied a process like squaring or adding that made the denominator bigger and bigger. This also indicates the sequence was computed from start to end, not the other way around.
Hypothesis: the generator separately tracks the numerator and denominator and uses the xₙ₊₁ = 2*sqrt(1 - xₙ) rule exactly when this will result in both the numerator and denominator remaining integers.
Here is a plot of denominator ( of the closest fraction with denominator < 1000,000,000)
This looks exactly what you would expect if you started with a number that happened to be a fraction, and applied a process like squaring or adding that made the denominator bigger and bigger. This also indicates the sequence was computed from start to end, not the other way around.
Well xn+1=−2√1−xn appears around as often.
If this were true, there must be something aiming towards simplicity. (A huge numerator + denominator are unlikely to be squares)