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.
The second relation never occurs when xₙ is the negation of the previous xₙ₋₁.
Furthermore, the second relation is always followed by xₙ₊₁ = -xₙ (i.e. there is never a “skipped pair” pattern break immediately following). This means that the skips are unlikely to be random.
It doesn’t look as if there are a lot of other such relationships to be found. There are a few probably-coincidental simple linear relationships between consecutive numbers very near the start. There are lots of y=−x, quite a lot of 4x=4−y2, one maybe-coincidental 14−x=2−y2, one maybe-coincidental x=−4(1−y)2, some 2x=1−y2, and I’m not seeing anything else among the first 400 pairs.
[EDITED to add:] But I have only looked at relationships where all the key numbers are powers of 2; maybe I should be considering things with 5s in as well. (Or maybe it’s 2s and 10s rather than 2s and 5s.)
x followed by -x is common.
cases where
1−xn=(xn+1/2)2are also common.
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)
The second relation never occurs when xₙ is the negation of the previous xₙ₋₁.
Furthermore, the second relation is always followed by xₙ₊₁ = -xₙ (i.e. there is never a “skipped pair” pattern break immediately following). This means that the skips are unlikely to be random.
Whenever 1−xn=(xn+1/2)2, this quantity is at most 4.
I’m finding also around 50 instances of 1−2xn=(xn+1)2∈[1,4] (namely 1≤|xn+1|≤2), with again xn+2=−xn+1.
It doesn’t look as if there are a lot of other such relationships to be found. There are a few probably-coincidental simple linear relationships between consecutive numbers very near the start. There are lots of y=−x, quite a lot of 4x=4−y2, one maybe-coincidental 14−x=2−y2, one maybe-coincidental x=−4(1−y)2, some 2x=1−y2, and I’m not seeing anything else among the first 400 pairs.
[EDITED to add:] But I have only looked at relationships where all the key numbers are powers of 2; maybe I should be considering things with 5s in as well. (Or maybe it’s 2s and 10s rather than 2s and 5s.)
There is one place where −xn=4(1−xn+1)2. I wonder whether there are many other relationships of broadly similar shape.