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Don Knuth found this [https://oeis.org/search?q=A220952 sequence] "so fascinating":
Don Knuth found this [https://oeis.org/search?q=A220952 sequence] so fascinating:
  0, 1, 2, 3, 4, 9, 14, 19, 18, 17, 16, 11, 12, 13, 8, 7, 6, 5, 10, 15, 20, 21, 22, 23, 24, 49
  0, 1, 2, 3, 4, 9, 14, 19, 18, 17, 16, 11, 12, 13, 8, 7, 6, 5, 10, 15, 20, 21, 22, 23, 24, 49
I stumbled over it when looking for OEIS sequences with keyword <code>unkn</code>.  
I stumbled over it when looking for OEIS sequences with keyword <code>unkn</code>.  
===Observations===
===Observations===
Here are the values listed in base 5:
The values listed in base 5 are: 00,01,02,03,04,14,24,34,33,32,31,21,22,23,13,12,11,10,20,30,40,41,42,43,44,144
n      a(n)  base 5
===================
0      0        0
-------------------     
1        1      1     
2        2      2     
3        3      3     
4        4      4
-------------------     
5        9      14    
6      14      24    
7      19      34    
8      18      33    
9      17      32    
10      16      31    
11      11      21    
12      12      22    
13      13      23    
14      8      13    
15      7      12    
16      6      11    
17      5      10     
18      10     20    
19      15      30    
20      20      40    
21      21      41    
22      22      42    
23      23      43    
24      24      44    
-------------------
25      49    144    
   
   
Some elementary conditions (disregarding the trailing 49):
Some elementary observations and conditions (disregarding the trailing 49):
* A permutation of the numbers 0..24
* a(n=0..24) is a permutation of the numbers 0..24
* a(n=0..12) = 24 - a(24-n)
* a(n=0..12) = 24 - a(24-n)
* Symmetricy around n=12, a(12) = 2*(5<sup>1</sup> + 5<sup>0</sup>)  
* Symmetricy around n=12, a(12) = 2*(5<sup>1</sup> + 5<sup>0</sup>)  
* Groups of length 5<sup>n</sup> - 1, n=0,1,2 ...
* Groups of length 5<sup>n</sup> - 1, n=0,1,2 ...
* Only one base-5 digit is incremented or decremented by 1 (without carry) in each step.


In the following I will refer to ''generations'' which correspond to the powers of 5. Knuth's values are generation 2 and the first of geneartion3 (49).
In the following I will refer to ''generations'' which correspond to the powers of 5. Knuth's values are generations 0,1,2 and the first of 3 (49).


I wrote a Perl program which plots the values of b-files with SVG. The [http://www.teherba.org/images/0/03/A220952.orig.svg image] shows a slighty distorted "Z" shape with "handles" on the diagonal. The nodes' values are showed in base 5. With another Perl program I tried to generate the starting 26 values, and a continuation up to the next power of 5 (125). The program did maintain the conditions noted above, but returned ten thousands of "unpleasant solutions". The conditions seem not to be strong enough.
I wrote a Perl program which plots the values of b-files with SVG. The [http://www.teherba.org/images/0/03/A220952.orig.svg image] shows a slighty distorted "Z" shape with "handles" on the diagonal. The nodes' values are showed in base 5.  
===Proposal===
===First Proposal===
Then I produced a big sheet of quad paper and tried to extend the basic shape to 125 elements. Please have a look at '''[http://www.teherba.org/images/7/71/Z53.svg my proposal]''' for a continuation with some "fractal" appearance.
Then I produced a big sheet of quad paper and tried to extend the basic shape to 125 elements. Please have a look at '''[http://www.teherba.org/images/7/71/Z53.svg my proposal]''' for a continuation with some "fractal" appearance.


I think that a(n=0..125) could read:
I thought that a(n=0..125) could read:
  1, 2,3,4,9, 14,19,18,17,16,11,12,13,8,7,6,5,10,15,20,21,22,23,24,  
  1, 2,3,4,9, 14,19,18,17,16,11,12,13,8,7,6,5,10,15,20,21,22,23,24,  
  49,74,99,98,97,96,95,90,85,80,81,82,83,88,87,86,91,92,93,94,89,84,79,78,77,76,51,
  49,74,99,98,97,96,95,90,85,80,81,82,83,88,87,86,91,92,93,94,89,84,79,78,77,76,51,
Line 55: Line 25:
  109,114,119,118,117,116,111,112,113,108,107,106,105,110,115,120,121,122,123,124,  
  109,114,119,118,117,116,111,112,113,108,107,106,105,110,115,120,121,122,123,124,  
  249
  249
I would have explained this proposal (for generation 3) as follows: For the "Z"s on the main diagonal, the upper, horizontal stroke consists of 4 nodes spanning a lenthagth of 3*5<sup>n</sup>. Before that upper stroke and behind the lower stroke there is a "handle" consisting of the shape of generation 2. Whenever the generation 2 shape occurs again in the sequence, it is mirrored on the vertical axis. Please note that the "shape" of generation 1 consists of the values 1,2,3 which are also visible - each time mirrored - on the "/"-stroke of generation 2.


===Explanation of Generation 3===
For generation 2, the program which evaluates the conditions above found 2 similiar variants:
For the "Z"s on the main diagonal, the upper, horizontal stroke consists of 4 nodes spanning a length of 3*5<sup>n</sup>. Before that upper stroke and behind the lower stroke there is a "handle" consisting of the shape of generation 2. Whenever the generation 2 shape occurs again in the squence, it is mirrored on the vertical axis. Please note that the "shape" of generation 1 consists of the values 1,2,3 which are also visible - each time mirrored - on the "/"-stroke of generation 2.
* 0, 1,2,3,4, 9,8,13,14,19,18,17,12,7,6,5,10,11,16,15,20,21,22,23,24 -  [http://www.teherba.org/images/6/64/Z53.var_a.svg a "Z" shape with broken horizontal strokes]
* 0, 1,2,3,4, 9,8,7,6,5,10,11,12,13,14,19,18,17,16,15,20,21,22,23,24 - [http://www.teherba.org/images/8/89/Z53.var_b.svg 2 small "Z" shapes]


Therefore I hope to be able to write a generating program for higher powers of 5. Bbut I have no idea for additional properties of this sequence.
===History of Sequence A220952===
Since sequence [https://oeis.org/search?q=A220952 A220952] still had the keyword '''unkn''' after 4 years, I wrote my proposal to the [http://list.seqfan.eu/ Seqfan Mailing list]. The discussion there showed quickly that:
* The problem had been stated by Donald Knuth in more detail in ''A twisted enumeration of the positive integers''; Problem 11733, Amer. Math. Monthly, 120 (9) (2013), 76.
* It was solved by Richard Stong in [https://lupucezar.files.wordpress.com/2011/02/amer-math-monthly-123-1-97.pdf Amer. Math. Monthly, 123 (1) (2016), 98-100].
The entry for the sequence was written by R. J. Mathar, together with a Maple program. Since I do not have a Maple license, I wrote [[OEIS/gen_meander.pl|Perl program]] which generates the same output.
 
===Why is it so fascinating?===
Maybe (also?) because it is a [https://en.wikipedia.org/wiki/FASS_curve FASS curve]. '''[[OEIS/FASS curves|Please read on]]''' to see what I have collected for these curves in the meantime.

Latest revision as of 03:25, 14 April 2019

Don Knuth found this sequence so fascinating: 

0, 1, 2, 3, 4, 9, 14, 19, 18, 17, 16, 11, 12, 13, 8, 7, 6, 5, 10, 15, 20, 21, 22, 23, 24, 49

I stumbled over it when looking for OEIS sequences with keyword unkn.

Observations

The values listed in base 5 are: 00,01,02,03,04,14,24,34,33,32,31,21,22,23,13,12,11,10,20,30,40,41,42,43,44,144

Some elementary observations and conditions (disregarding the trailing 49):

  • a(n=0..24) is a permutation of the numbers 0..24
  • a(n=0..12) = 24 - a(24-n)
  • Symmetricy around n=12, a(12) = 2*(51 + 50)
  • Groups of length 5n - 1, n=0,1,2 ...
  • Only one base-5 digit is incremented or decremented by 1 (without carry) in each step.

In the following I will refer to generations which correspond to the powers of 5. Knuth's values are generations 0,1,2 and the first of 3 (49).

I wrote a Perl program which plots the values of b-files with SVG. The image shows a slighty distorted "Z" shape with "handles" on the diagonal. The nodes' values are showed in base 5.

First Proposal

Then I produced a big sheet of quad paper and tried to extend the basic shape to 125 elements. Please have a look at my proposal for a continuation with some "fractal" appearance.

I thought that a(n=0..125) could read: 

1, 2,3,4,9, 14,19,18,17,16,11,12,13,8,7,6,5,10,15,20,21,22,23,24, 
49,74,99,98,97,96,95,90,85,80,81,82,83,88,87,86,91,92,93,94,89,84,79,78,77,76,51,
52,53,54,59,64,69,68,67,66,61,62,63,58,57,56,55,60,65,70,71,72,73,48,47,46,45,40,
35,30,31,32,33,38,37,36,41,42,43,44,39,34,29,28,27,26,25,50,75,100,101,102,103,104,
109,114,119,118,117,116,111,112,113,108,107,106,105,110,115,120,121,122,123,124, 
249

I would have explained this proposal (for generation 3) as follows: For the "Z"s on the main diagonal, the upper, horizontal stroke consists of 4 nodes spanning a lenthagth of 3*5n. Before that upper stroke and behind the lower stroke there is a "handle" consisting of the shape of generation 2. Whenever the generation 2 shape occurs again in the sequence, it is mirrored on the vertical axis. Please note that the "shape" of generation 1 consists of the values 1,2,3 which are also visible - each time mirrored - on the "/"-stroke of generation 2.

For generation 2, the program which evaluates the conditions above found 2 similiar variants:

History of Sequence A220952

Since sequence A220952 still had the keyword unkn after 4 years, I wrote my proposal to the Seqfan Mailing list. The discussion there showed quickly that:

  • The problem had been stated by Donald Knuth in more detail in A twisted enumeration of the positive integers; Problem 11733, Amer. Math. Monthly, 120 (9) (2013), 76.
  • It was solved by Richard Stong in Amer. Math. Monthly, 123 (1) (2016), 98-100.

The entry for the sequence was written by R. J. Mathar, together with a Maple program. Since I do not have a Maple license, I wrote Perl program which generates the same output.

Why is it so fascinating?

Maybe (also?) because it is a FASS curve. Please read on to see what I have collected for these curves in the meantime.

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