Runs2

(which self-describe their sizes)

Look at A:

A = 1,10,3,4,5,6,7,8,9,11,21,12,31,13,14,110,15,41,16,17,18,112,19,20,51,100,101,201,102,22,23,24,211,...

Now insert a vertical stroke between all consecutive 1’s:

A = 1,10,3,4,5,6,7,8,9,11, 21,12,31,13,14,110, 15,41,16,17,18,112, 19,20,51,100,101,201,102,22,23,24,211,...

A = 1|10,3,4,5,6,7,8,9,1|1,21|12,31|13,14,1|10,15,41|16,17,18,1|12,19,20,51|100,101,201|102,22,23,24,21|1,...

The quantity of digits between two strokes is given by A itself:

A = 1|10,3,4,5,6,7,8,9,1|1,21|12,31|13,14,1|10,15,41|16,17,18,1|12,19,20,51|100,101,201|102,22,23,24,21|1,...

1 10 3 4 5 6 7 8 9 11

A is the lexicographically first such sequence; to build it we always use the smallest available integer not yet present in A and not leading to a contradiction.

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Look at B:

B = 3,1,22,20,4,5,6,7,8,9,10,11,12,13,14,15,122,16,17,18,19,21,23,24,25,26,220,322,30,32,27,28,42,29,...

We insert now a vertical stroke between all consecutive 2’s:

B = 3,1,22, 20,4,5,6,7,8,9,10,11,12,13,14,15,122, 16,17,18,19,21,23,24,25,26,220, 322, 30,32,27,28,42,29,...

B = 3,1,2|2|20,4,5,6,7,8,9,10,11,12,13,14,15,12|2,16,17,18,19,21,23,24,25,26,2|20,32|2,30,32|27,28,42|29,...

Again, we see that the quantity of digits between two strokes is given by B itself:

B = 3,1,2|2|20,4,5,6,7,8,9,10,11,12,13,14,15,12|2,16,17,18,19,21,23,24,25,26,2|20,32|2,30,32|27,28,42|29,...

3 1 22 20 4 5 6

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Now C:

C = 2,3,30,33,1,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,23,31,20,21,22,24,25,26,27,28,29,32,34,35,36,37,333,40,330,39,331,41,43,300,40,53,301,...

We insert our vertical strokes between all consecutive 3’s:

C = 2,3,30,33, 1,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,23,31,20,21,22,24,25,26,27,28,29,32,34,35,36,37,38,333, 40,330, 39,331, 41,43,300,40,53,301,...

C = 2,3|30,3|3,1,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,23|31,20,21,22,24,25,26,27,28,29,32,34,35,36,37,38,3|3|3,40,3|30,39,3|31,41,43|300,40,53|301,...

As usual, the quantity of digits between two strokes is given by C itself:

C = 2,3|30,3|3,1,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,23|31,20,21,22,24,25,26,27,28,29,32,34,35,36,37,38,3|3|3,40,3|30,39,3|31,41,43|300,40,53|301,...

2 3 30 33 1 4 5 6 7

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Etc.

Could someone be so kind to compute a hundred terms or so for A, B, C and the remaining sequences:

D = 4|4

E = 5|5

F = 6|6

G = 7|7

H = 8|8

I = 9|9

J = 0|0

Best,

É.

(same flavor here)