Periodic Strings

[Antoni Gual]

Antoni states:
Starting point=11606
Period=8689

String fragment at 11606:
33333333312526283231323025211925273434394037374037343839404136383836414239403329353836424338413533333

String fragment at 20295 (11606+8689):
33333333327302730273125302326242728364036394237374443374238384140333430211926283133373637364140343227

These do not match.

Obviously my way of finding repeats is not at fault. both strings start by 9 threes, so the next values should be equal. As they aren't, it is my way of generating the series what is at fault. One of both series of nine threes must be incorrect....

Quibbler: strings are not slow, provided you never resize the big ones... Try it!!

[whodat]

Antoni states:
Starting point=11606
Period=8689

String fragment at 11606:
33333333312526283231323025211925273434394037374037343839404136383836414239403329353836424338413533333

String fragment at 20295 (11606+8689):
33333333327302730273125302326242728364036394237374443374238384140333430211926283133373637364140343227

These do not match.

Obviously my way of finding repeats is not at fault. both strings start by 9 threes, so the next values should be equal. As they aren't, it is my way of generating the series what is at fault. One of both series of nine threes must be incorrect....

Quibbler: strings are not slow, provided you never resize the big ones... Try it!!

Both strings start with 9 threes, but the next values don't necessarily have to be equal. Remember, these are fragments pulled from the middle of the string at your recommended points.

The second string at 20295 has 9 threes giving a sum of 27, but in the first string at 11606 it is 8 threes plus 1 giving a sum of 25.
There is no contradiction here.

[Antoni Gual]

I see. The existance of multi- figure results breaks my algorithm...

[Agamemnus]

Very interesting.

We know one thing immediately: all strings must repeat at some point because there is a limit of how many numbers can be cycled through.

The general equation is such:

A(1) through A(m) = 1

Starting with A(m+1),
A(n) through A(n+D) = digits [where digits = D] of:
sum(A(k)) where k = n-1 through n-m.


A(X) through A(X+Z) = A(X+Z+1) through A(X+2Z) = A(X+2Z + 1) through A(X+3Z) ...

Find the formula for X, Z, and Y for any "m".

-----------------------------------------


One solution is to pre-generate a very large string, then convert each string to 9 separate strings. Each one would either contain the number 0-9 or "-". Then convert that to frequencies of that number or "-". Drop the last empties.

Code:

Example, arbitrarily set end to 25:
1123581347 1123581347 11235 becomes:
11----1--- 11----1--- 11---
--2------- --2------- --2--
---3---3-- ---3---3-- ---3-
----5----- ----5----- ----5
-----8---- -----8---- -----
--------4- --------4- -----
---------7 ---------7 -----

1: 241324132-
2: 219191-
3: 3131513151-
4: 7191-
5: 4159191
6:
7: 9191-
8: 51491-
9:

Skip the empty ones.

You can generate this list automatically as you go. Apply your basic Gual repeating sequence finder to all of these new ones. If all of them are found to be repeating, you can stop and figure out what the big sequence is.. I haven't figured out how to do that bit though. It seems so simple but I just can't figure out exactly how to do it.

You can make the number sequence go in hundreds or thousands instead as well.

You can also get the sequences inside THESE sequences... and so on...

[whodat]

I finally got my QB program to work for the 9th order string. The period for this one seemed to be very long, and I kept getting "out of string space" errors. Instead I saved the data to a file (200k), extracted the last 255 digits, and looked for matches starting at the beginning of the file. The distance between the first two matches gave me the period length.

Knowing the period length, I again went back to the beginning of the file, and looked for the first two repeats of that period. The start of the first repeat gave me the start of periodicity.

Results:
starting point is 67925
period length is 24405

These results agree with those first reported by Quibbler on Oct. 21

[Agamemnus]

Ok, I figured out how to combine the sequences... just sort them depending on what location each sequence started..