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.