The first few numbers are mostly “nice”. 1.3, 2.1, −0.5, 0.65, 1.05, 0.675, −1.15, 1.15, 1.70875, 0.375, −0.375, −0.3225, −2.3, 2.3. Next after that is 1.64078125, which is not so nice but still pretty nice.
These are 13⁄10, 21⁄10, −1/2, 13⁄20, 21⁄20, 27⁄40, −23/20, 23⁄20, 1367⁄800, 3⁄8, −3/8, −129/400, −23/10, 23⁄10, 10501⁄6400, …
A few obvious remarks about these: denominators are all powers of 2 times powers of 5; 13 and 21 are Fibonacci numbers; shortly after 13⁄10 and 21⁄10 we have 13⁄20 and 21⁄20; we have three cases of x followed by -x.
Slightly less than 1⁄3 of all numbers are followed by their negatives. Usually it goes a, -a, b, c, -c, d, etc., but (1) the first x,-x pair isn’t until the 7th/8th numbers, and (2) sometimes there is a longer gap and (3) there are a few places where there’s a shorter gap and we go x, -x, y, -y.
Most of the breaks in the (a, -a, b, c, -c, d) pattern look like either the +/- pair was skipped entirely or the unpaired value was skipped. My guess is the complete message consists of alternating “packets” of either a single value or a +/- pair, and each packet has some chance to be omitted (or they are deterministically omitted according to some pattern).
The number 23⁄20 appears three times. Each time it appears it is followed by a different number, and that different number is generally “unusually nasty compared with others so far”.
First time, at (0-based) index 7: followed by 1367⁄800, first with a denominator > 40.
Second time, at (0-based) index 21: followed by 3.1883919921875 ~= 1632.4567/2^9, first with a denominator > 6400.
Third time, at (0-based) index 48: followed by 2.4439140274654036, dunno what this “really” is, first with no obvious powers-of-2-and-5 denominator.
If we multiply it by 2^11/1001, the number we actually get is 5.00013579245669; that decimal tail also seems just slightly suspicious. 1, 3, 5, 7, 9, 2, 4, 6, approximately-8.
Early in the sequence (i.e., before roundoff error has much effect, if there’s something iterative going on) it seems like a surprising number of our numbers have denominators that are (power of 2) x 10000. As if there’s something happening to 4 decimal places, alongside something happening that involves exact powers of 2. (Cf. Measure’s conjecture that something is keeping track of numerators and denominators separately.) This is all super-handwavy and I don’t have a really concrete hypothesis to offer.
[EDITED to add:] The apparent accumulating roundoff is itself evidence against this, because it means that after a while our numbers are not powers of 2 over 10000. So I’ll be quite surprised if this turns out to be anything other than delusion. I’m leaving it here just in case it gives anyone useful ideas.
These aren’t all quite correctly rounded. E.g., the 12th number is about −0.3225 but it isn’t the nearest IEEE754 doublefloat to −0.3225. I suspect these are “honest” rounding errors (i.e., the values were produced by some sort of computation with roundoff error) rather than there being extra hidden information lurking in the low bits.
The first few numbers are mostly “nice”. 1.3, 2.1, −0.5, 0.65, 1.05, 0.675, −1.15, 1.15, 1.70875, 0.375, −0.375, −0.3225, −2.3, 2.3. Next after that is 1.64078125, which is not so nice but still pretty nice.
These are 13⁄10, 21⁄10, −1/2, 13⁄20, 21⁄20, 27⁄40, −23/20, 23⁄20, 1367⁄800, 3⁄8, −3/8, −129/400, −23/10, 23⁄10, 10501⁄6400, …
A few obvious remarks about these: denominators are all powers of 2 times powers of 5; 13 and 21 are Fibonacci numbers; shortly after 13⁄10 and 21⁄10 we have 13⁄20 and 21⁄20; we have three cases of x followed by -x.
Slightly less than 1⁄3 of all numbers are followed by their negatives. Usually it goes a, -a, b, c, -c, d, etc., but (1) the first x,-x pair isn’t until the 7th/8th numbers, and (2) sometimes there is a longer gap and (3) there are a few places where there’s a shorter gap and we go x, -x, y, -y.
Most of the breaks in the (a, -a, b, c, -c, d) pattern look like either the +/- pair was skipped entirely or the unpaired value was skipped. My guess is the complete message consists of alternating “packets” of either a single value or a +/- pair, and each packet has some chance to be omitted (or they are deterministically omitted according to some pattern).
The number 23⁄20 appears three times. Each time it appears it is followed by a different number, and that different number is generally “unusually nasty compared with others so far”.
First time, at (0-based) index 7: followed by 1367⁄800, first with a denominator > 40.
Second time, at (0-based) index 21: followed by 3.1883919921875 ~= 1632.4567/2^9, first with a denominator > 6400.
Third time, at (0-based) index 48: followed by 2.4439140274654036, dunno what this “really” is, first with no obvious powers-of-2-and-5 denominator.
[EDITED to fix an error.]
2.4439140274654036 might be (3³x19×3671×10631)/(2¹⁹x5⁶) with some incorrect rounding (2.4439140274658203125).
Value[71] is exactly half of value[49]. (and this again follows a 23)
The ”.4567″ seems just slightly suspicious.
That third number is quite close to 5005/2^11.
If we multiply it by 2^11/1001, the number we actually get is 5.00013579245669; that decimal tail also seems just slightly suspicious. 1, 3, 5, 7, 9, 2, 4, 6, approximately-8.
This could all be mere pareidolia, obviously.
Early in the sequence (i.e., before roundoff error has much effect, if there’s something iterative going on) it seems like a surprising number of our numbers have denominators that are (power of 2) x 10000. As if there’s something happening to 4 decimal places, alongside something happening that involves exact powers of 2. (Cf. Measure’s conjecture that something is keeping track of numerators and denominators separately.) This is all super-handwavy and I don’t have a really concrete hypothesis to offer.
[EDITED to add:] The apparent accumulating roundoff is itself evidence against this, because it means that after a while our numbers are not powers of 2 over 10000. So I’ll be quite surprised if this turns out to be anything other than delusion. I’m leaving it here just in case it gives anyone useful ideas.
These aren’t all quite correctly rounded. E.g., the 12th number is about −0.3225 but it isn’t the nearest IEEE754 doublefloat to −0.3225. I suspect these are “honest” rounding errors (i.e., the values were produced by some sort of computation with roundoff error) rather than there being extra hidden information lurking in the low bits.