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. Here are the values listed in 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 (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 ...

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. 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 would maintain the conditions noted above, but there are ten thousands of - ugly - solutions.

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.