OEIS/A220952

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

Here are the values listed in base 10 and base 5:

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 observations and conditions (disregarding the trailing 49):

• 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:

• 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 - 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 - 2 small "Z" shapes

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 why I am fascinated in the meantime.