For what it’s worth, colored by how soon in the sequence they appear (blue is early, red is late) (Also note I interpreted it as 2094 points, with each number first used in the x-dimension and then in the y-dimension):
Note that one line near the top appears to be drawn twice, confirming if nothing else that it’s not a rule that it’s not a succession rule that only depends on the previous value, since the paths diverge afterwards. Still, comparing those two sections could be interesting.
The double line I was talking about is actually a triple line, at indices 366, 677, and 1244. The lines before come from fairly different places, and they diverge pretty quickly afterwards:
However, just above it, there’s another duplicate line, at indices 1038 and 1901:
These start out closer together and also take a little bit longer to diverge.
This might be indicative of a larger pattern that points that are close together and have similar histories tend to have their next steps close to each other as well.
Every number in the sequence which equals 1-(y/2)^2 for “nice” y to “good” accuracy is in fact on the quadratic; i.e., when a number is “approximately” 1-(y/2)^2 for a y that “could” come next, that y does in fact come next. BUT to make this so I am having to define “nice” and “good” fairly tightly; this in fact occurs for only 14 values early in the sequence, perhaps before roundoff error starts being a nuisance.
(I am conjecturing that the way this thing is constructed might be that we start with a few initial values and then iterate some process like: “if (condition) then next number is—previous number; else if (condition) then next number is 2sqrt(1-previous); else if …”. If so, then we might start with “nice” numbers—in fact the first two are 1.3 and 2.1 to machine precision—but that “niceness” might be eroded as we calculate. This is all very handwavy and there is at least one kinda-obvious objection to it.)
A few reasons for suspecting that this sort of iterative process is in play:
It would fit with the relationship to “That Alien Message” where the idea is that each frame (here, each number) is the “next” state of affairs as something evolves according to simple rules.
There are many cases in which it sure looks as if one number is derived from its predecessor. (Or maybe from its successor.)
It’s an obvious thing to do, if you want to set a challenge like this.
Note that I am not necessarily conjecturing that each number is a function of its predecessor. There might be other internal state that we don’t get to see.
Do you see how such an iteration can produce the long-distance correlations I mention in a message below, between floats at positions that differ by a factor of 2/e? It seems that this would require some explicit dependence on the index.
Its not exactly 2/e. Here is a plot of the “error” of those points. The x axis is the larger point. The y axis is the smaller point minus 2/e times the larger.
So its within about 1% of 2/e, suggesting it might be a real thing, or might just be a coincidence.
(I am conjecturing that the way this thing is constructed might be that we start with a few initial values and then iterate some process like: “if (condition) then next number is—previous number; else if (condition) then next number is 2sqrt(1-previous); else if …”. If so, then we might start with “nice” numbers—in fact the first two are 1.3 and 2.1 to machine precision—but that “niceness” might be eroded as we calculate. This is all very handwavy and there is at least one kinda-obvious objection to it.)
Have you tried writing programs that create sequences like the above to see how close they get? (Asking to avoid redoing work if the answer is yes.)
Making explicit something I said in passing elsewhere: it seems not-impossible that the sequence might be best looked at in reverse order, since mapping x to 1-(x/2)^2 or (1-x^2)/2 is somewhat more natural than going the other way. However, it does look rather as if there might be some accumulation of roundoff error in the forward direction, which if true would argue for reading it forwards rather than backwards.
More structure emerges! Here’s a plot of consecutive pairs of values (data[i], data[i+1]) such that data[i+1] = -data[i+2]. 
Has anyone tried a five-dimensional representation instead of a two-dimensional one? 2095 isn’t divisible by 2 or by 3, but it is divisible by 5. Maybe our “aliens” have four spatial dimensions and one temporal.
I’ve tried highlighting every k-th point out of n with the same color for a bunch of n, but it all looks random. Right now, I’ve also tried using only 2 of 5 float values. It looks like a dead end, even though the idea is good.
I think the data is 1-dimensional, the interesting part is how each number is transformed into the next, and the 2d representation just happens to catch that (e.g., if x is transformed into −x, it lands on the diagonal).
Second try: When looking at scatterplots of any 3 out of 5 of those dimensions and interpreting each 5-tuple of numbers as one point, you can see the same structures that are visible in the 2d plot, the parabola and a line—though the line becomes a plane if viewed from a different angle, and the parabola disappears if viewed from a different angle
The straight line comes from the (x,-x) pairs I remarked on, which make up ~2/3 of the dataset. The parabolic thing comes from values …,a,b,… where a = 1-(b/2)^2 to something like full precision.
Here’s what you get when you interpret them as (y,x) positions of points.
For what it’s worth, colored by how soon in the sequence they appear (blue is early, red is late) (Also note I interpreted it as 2094 points, with each number first used in the x-dimension and then in the y-dimension):
Note that one line near the top appears to be drawn twice, confirming if nothing else that it’s not a rule that it’s not a succession rule that only depends on the previous value, since the paths diverge afterwards.
Still, comparing those two sections could be interesting.
The double line I was talking about is actually a triple line, at indices 366, 677, and 1244. The lines before come from fairly different places, and they diverge pretty quickly afterwards:
However, just above it, there’s another duplicate line, at indices 1038 and 1901:
These start out closer together and also take a little bit longer to diverge.
This might be indicative of a larger pattern that points that are close together and have similar histories tend to have their next steps close to each other as well.
Every number in the sequence which equals 1-(y/2)^2 for “nice” y to “good” accuracy is in fact on the quadratic; i.e., when a number is “approximately” 1-(y/2)^2 for a y that “could” come next, that y does in fact come next. BUT to make this so I am having to define “nice” and “good” fairly tightly; this in fact occurs for only 14 values early in the sequence, perhaps before roundoff error starts being a nuisance.
(I am conjecturing that the way this thing is constructed might be that we start with a few initial values and then iterate some process like: “if (condition) then next number is—previous number; else if (condition) then next number is 2sqrt(1-previous); else if …”. If so, then we might start with “nice” numbers—in fact the first two are 1.3 and 2.1 to machine precision—but that “niceness” might be eroded as we calculate. This is all very handwavy and there is at least one kinda-obvious objection to it.)
A few reasons for suspecting that this sort of iterative process is in play:
It would fit with the relationship to “That Alien Message” where the idea is that each frame (here, each number) is the “next” state of affairs as something evolves according to simple rules.
There are many cases in which it sure looks as if one number is derived from its predecessor. (Or maybe from its successor.)
It’s an obvious thing to do, if you want to set a challenge like this.
Note that I am not necessarily conjecturing that each number is a function of its predecessor. There might be other internal state that we don’t get to see.
Do you see how such an iteration can produce the long-distance correlations I mention in a message below, between floats at positions that differ by a factor of 2/e? It seems that this would require some explicit dependence on the index.
Its not exactly 2/e. Here is a plot of the “error” of those points. The x axis is the larger point. The y axis is the smaller point minus 2/e times the larger.
So its within about 1% of 2/e, suggesting it might be a real thing, or might just be a coincidence.
Have you tried writing programs that create sequences like the above to see how close they get? (Asking to avoid redoing work if the answer is yes.)
Making explicit something I said in passing elsewhere: it seems not-impossible that the sequence might be best looked at in reverse order, since mapping x to 1-(x/2)^2 or (1-x^2)/2 is somewhat more natural than going the other way. However, it does look rather as if there might be some accumulation of roundoff error in the forward direction, which if true would argue for reading it forwards rather than backwards.
More structure emerges! Here’s a plot of consecutive pairs of values (data[i], data[i+1]) such that data[i+1] = -data[i+2]. 
Has anyone tried a five-dimensional representation instead of a two-dimensional one? 2095 isn’t divisible by 2 or by 3, but it is divisible by 5. Maybe our “aliens” have four spatial dimensions and one temporal.
I’ve tried highlighting every k-th point out of n with the same color for a bunch of n, but it all looks random. Right now, I’ve also tried using only 2 of 5 float values. It looks like a dead end, even though the idea is good.
I think the data is 1-dimensional, the interesting part is how each number is transformed into the next, and the 2d representation just happens to catch that (e.g., if x is transformed into −x, it lands on the diagonal).
Second try: When looking at scatterplots of any 3 out of 5 of those dimensions and interpreting each 5-tuple of numbers as one point, you can see the same structures that are visible in the 2d plot, the parabola and a line—though the line becomes a plane if viewed from a different angle, and the parabola disappears if viewed from a different angle
Looking at scatterplots of any 3 out of 5 of those dimensions, it looks pretty random, much less structure than in the 2d plot.
Edit: Oh, wait, I’ve been using chunks of 419 numbers as the dimensions but should be interleaving them
(This has y increasing downwards.)
The straight line comes from the (x,-x) pairs I remarked on, which make up ~2/3 of the dataset. The parabolic thing comes from values …,a,b,… where a = 1-(b/2)^2 to something like full precision.
Ah, it’s the Elden Ring
There are 1047 points. The points on the diagonal are distributed all across the list—here are the first indices
[3,6,9,10,13,16,18,21,25,27,28,31,34,38,41,43,46,49,52,55]...
They’re usually 3 apart, but sometimes 1, 2, 4, 5, or 6. The last one has index 1045.