Here's an execution for 2, 3. When Furthermore, note that if therefore we can always make
a != 1 we use a as the extra numbers instead of b: 1 | 2 2(1) ...
2 | 2 2(0) 2(1) ...
3 | 3 2(1) 2(0) 2(1) ...
4 | 3 2(0) 2(0) 2(1) ...
5 | 2 3(2) 2(1) 2(0) 2(0) ...
6 | 2 3(1) 2(1) 2(0) 2(1) ...
7 | 2 3(0) 2(1) 2(0) 2(1) ...
8 | 3 3(2) 2(0) 2(0) 2(1) ...
9 | 3 3(1) 2(0) 2(0) 2(1) ...
10 | 3 3(0) 2(0) 2(0) 2(1) ...
11 | 2 2(1) 3(2) 2(1) 2(0) 2(1) ...
12 | 2 2(0) 3(2) 2(1) 2(0) 2(1) ...
13 | 3 2(1) 3(1) 2(1) 2(0) 2(1) ...
14 | 3 2(0) 3(1) 2(1) 2(0) 2(1) ...
15 | 2 2(1) 3(0) 2(1) 2(0) 2(1) ...
16 | 2 2(0) 3(0) 2(1) 2(0) 2(1) ...
17 | 3 3(2) 3(2) 2(0) 2(0) 2(1) ...a = 1, then the a, b sequence is a subset of the b, a sequence e.g.:1, 2 = [1, 2, 2, 1, 1, 2, 1, ...]
2, 1 = [ 2, 2, 1, 1, 2, 1, ...]a not be 1 by switching the pair and then using the generalized algorithm with a != 1.This algorithm is more efficient in space, using only , as it recursively compresses the state required to keep track of what to do next.
Time is still .
The table at maths-people.anu.edu.au/~brent/pd/Kolakoski-UNSW.pdf page 20 has a summary image, but it is hard to understand.
1 2(0) 1(1)The starting state is:which means that it implicitly contains infinitely many The actual algorithm will of course omit as many trailing
2 | 2 2(1) 2(1) 2(1) 2(1) ...2(1) at the end which we abbreviate just as:2 | 2 2(1) ...2(1) as it can.The update rules are:Note that both rules don't overlap so that each update is always determined by only one of them at a time.
- go left to right:
- flip:continue going left to right.
x(0) y(0) !x((!x)-1) unchanged - repeat:and then stop further updates.
x(0) y(n > 0) x(x-1) y(n - 1)
- flip:
Also the first column is always implicitly
(0).2 | 2 2(1) ...
3 | 2 2(0) 2(1) ...Here we:
2 | 2 2(1) ...
3 | 2 2(0) 2(1) ...
4 | 1 2(1) 2(0) 2(1) ...2 | 2 2(1) ...
3 | 2 2(0) 2(1) ...
4 | 1 2(1) 2(0) 2(1) ...
5 | 1 2(0) 2(1) 2(0) 2(1) ... 2 | 2 2(1) ...
3 | 2 2(0) 2(1) ...
4 | 1 2(1) 2(0) 2(1) ...
5 | 1 2(0) 2(0) 2(1) ...
6 | 2 1(0) 2(1) 2(0) 2(1) ...
7 | 1 1(0) 2(0) 2(0) 2(1) ...
8 | 2 2(1) 1(0) 2(1) 2(0) 2(1) ...
9 | 2 2(0) 1(0) 2(1) 2(0) 2(1) ...
10 | 1 1(0) 1(0) 2(0) 2(0) 2(1) ...
11 | 2 2(1) 2(1) 1(0) 2(1) 2(0) 2(1) ...
12 | 2 2(0) 2(1) 1(0) 2(1) 2(0) 2(1) ...
13 | 1 2(1) 2(0) 1(0) 2(1) 2(0) 2(1) ...
14 | 1 2(0) 2(0) 1(0) 2(1) 2(0) 2(1) ...
15 | 2 1(0) 1(0) 1(0) 2(0) 2(0) 2(1) ...
16 | 1 2(1) 2(1) 2(1) 1(0) 2(1) 2(0) 2(1) ...The generalized Kolakoski sequence is the generalization of the Kolakoski sequence where you don't need to restrict yourself to 1,2 but can instead use any a,b pair.
What would be really amazing is if they had constraints like proper CAD software, it would make it possible to not have to redo entire diagrams when you want to change a small part of them.
Bibliography:
- gitlab.com/inkscape/inbox/-/issues/1465 Feature Request: Geometric Construction Tool (Constraints and constraint solver) for CAD, CAM etc.
- www.reddit.com/r/Inkscape/comments/1fvhil4/constraints_or_parameters/
It should be noted however that path effects can accomplish some of it.
There is a Notable section of I, Claudius that stuck with Ciro Santilli:
The idea of writing the history pleased me very much: I felt that historical justice had never been done to the Carthaginians. I spent my leisure time in making a study of the ruins of the Old City, with the help of a contemporary survey, and familiarizing myself with the geography of the country in general. I also learned the rudiments of the language well enough to be able to read simple inscriptions and understand the few Phoenician words used by authors who have written about the Punic Wars from the Roman side. When I returned to Italy I began to write the book concurrently with my Etruscan history. I like having two tasks going at the same time: when I tire of one I turn to the other. But I am perhaps too careful a writer. I am not satisfied merely with copying from ancient authorities while there is any possible means of checking their statements by consulting other sources of information on the same subject, particularly accounts by writers of rival political parties. So these two histories, each of which I could have written in a year or two if I had been less conscientious, kept me busy between them for some twenty-five years. For every word I wrote I must have read many hundreds; and in the end I became a very good scholar both of Etruscan and Phoenician, and had a working knowledge of several other languages and dialects too, such as Numidian, Egyptian, Oscan, and Faliscan. I finished the History of Carthage first.
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