Non-monotonic self-1-describing sequence

[Eric Angelini]

More on the same subject,

This is a non-monotonic "self-1-describing" sequence. It says :
« Consider the sequence as a succession of *digits*; then a(n)
indicates the position of a "1" in the sequence. All "1"s in 
the sequence are such described »
Additional building rule : no twice the same integer can appear
in the sequence; whenever there is a choice, put the smallest
available integer. This should give, if I'm not wrong:

1,3,10,6,11,7,21,13,15,17,19,101,24,100,29,102,34,103,39,104,44
105,49,106,54,107,59,108,64,109,69,110,70,76,111,77,78,85,112,
86,113,90,91,97,1111,11111,111115...

Those seq. are a great fun to compute by hand -- they are full
of surprises, you have to think backwards sometimes, and some-
times a long way upwards...

The above seq. starts like that:

1 (meaning « there is a "1" digit in position 1 in the seq. » --
   which is true, obviously);

What next?
   - "2" is forbidden (it would mean « there is a "1" digit in 
   position 2 in the seq. » -- which is false, because it's a "2"!)
   - "3" is ok -- it lets the "future" of the seq. open and forces
   the third digit; so we have:

1,3,1 

What next?
   - not a comma -- this would mean that the integer "1" is pre-
     sent twice in the seq., which is forbidden; so let us try
     the smallest two-digit available integer:

1,3,10,

This implies that the 10th digit of the sequence is a "1":

1,3,10,abcde1

What next?
    - the smallest available integer not used yet & not leading
      to a contradiction, is not "2", nor "3", "4" or "5" but "6";
      this "6" forces the 6th position of a "1":

1,3,10,6,1cde1

What next?
    - we cannot replace "c" with a "0" -- we have used already the
      integer "10"; so let's try integer "11" (this "11" forces a
       "1" in position 11):

1,3,10,6,11,de11

In chosing the integer "11" we have added a not-yet-described "1",
the one just before the d-letter; can we describe it right now? Yes,
this "1" is in 7th position, so "d" will be = 7:

1,3,10,6,11,7,e11

What next?
    - "e" cannot figure a single integer -- 8 or 9 -- as this would
      lead to a contradiction: there are no "1"s in position 8 or 9.
      Thus our next integer must be a two-digit one; not "11" (al-
      ready used) but 21 (smallest available, as the rule says); 
      this 21 forces a "1" in position 21; we have:

1,3,10,6,11,7,21,1abcdefghi1

What next?
    - "a" must be linked to his neighboring "1" to form a two-digit
      integer -- but neither 10 or 11 (already in the sequence); 12
      would lead to a self-contradiction (the 12th digit of the seq
      would be the "2" of "12"), so let's try 13, which forces a "1"
      in position 13, as usual:

1,3,10,6,11,7,21,13,1cdefghi1

What next?
    - "c" cannot be "4" ("14" would self-contradict again); "15" 
      will produce:

1,3,10,6,11,7,21,13,15,1efghi1

What next?
    - "e" cannot, etc. "17" produces:

1,3,10,6,11,7,21,13,15,17,1ghi1

What next?
    - "g" cannot, etc. "19" produces:

1,3,10,6,11,7,21,13,15,17,19,1i1

What next?
    - "i" must be "0" to form the smallest possible next term, together
      with both its neighbors -- and this is "101"; this "101" forces
      a "1" digit in position 101 in the seq. (not represented here --
      but this can be checked in the "full" above sequence):

1,3,10,6,11,7,21,13,15,17,19,101,

What next?
    - all "1"s so far in the sequence are (self-)described, so the next
      integer will describe the "future"; the smallest available and not
      self-contradictory integer is "24", which produces: 

1,3,10,6,11,7,21,13,15,17,19,101,24,1

What next?
    - this lone "1" cannot produce anything smaller than "100", forcing
      a "1" in 100th position (not represented here, but, etc.); we have:

1,3,10,6,11,7,21,13,15,17,19,101,24,100,

etc.

Who will start a « non-monotonic self-2-describing sequence » ;-?

Best,
É.