Add and write sequence

[Eric Angelini]

Hello Math-Fun and SeqFan,

Start with 1, then a(n) is the smallest integer not used so far
whose leftmost digits show the sum of a(n-1)'s digits:

1,10,11,2,20,21,3,30,31,4,40,41,5,50,51,6,60,61,7,70,71,8,80,81,
9,90,91,100,12,32,52,72,92,110,22,42,62,82,101,23,53,83,111,33,
63,93,120...

This infinite seq. is certainly a rearrangement of the natural 
integers.

Starting with my birthday year produces:

1951,16,7,70,71,8,80,81,9,90,91,10,1,11,2,20,21,...

The present year gives:

2007,9,90,91,10,1,11,2,20,21,3,30,31,...

Will the 1951-sequence and the 2007-sequence merge with the first
one, sooner or later?

This is the 11-sequence:

11,2,20,21,3,30,31,4,40,41,5,50,51,6,60,61,7,70,71,8,80,81,9,90,
91,10,1,12...

12 is there because 11 is forbidden; we could thus assign to 11 a 
"pseudo-loop" length -- the number of integers which separate the 
starting "11" from his expected second appearance. Here, this
pseudo-loop has length 27.

Could someone compute a few pseudo-loop lenghtes (say for n=1 to
n=200)?

This seq. obviously begins like this (n=1 to n=11):

1,1,1,1,1,1,1,1,1,2,27...

Best,
É.