[seqfan] Seq and First diff use only a 3-digit alphabet

[Eric Angelini]

Hello SeqFans,
here is the idea:

  seq Q = 1 9 18 19  118 119   1118 1119   11118 11119 ...
1st diff=  8 9  1  99   1   999    1    9999    1

... we use only the digits 1, 8 and 9 for Q and Q's 1st diff.
The pattern is ovious -- and even more obvious in Q':

  seq Q'= 8 9 18 19  118 119   1118 1119   11118 11119 ...
1st diff=  1 9  1  99   1   999    1    9999    1

Terms of Q and Q' are positive and monotonically increasing;
both seq are infinite.

R isn't:

  seq R = 3 7  44 47   END
1st diff=  4 37  3   ?

For a 2-digit alphabet, we have:

   T(1)=1,11,111,1111,11111,...
   T(2)=2,22,222,2222,22222,...
   ...
   T(9)=9,99,999,9999,99999,...

Adding a 3rd term to some T's seq above brings a bunch of new
3-digit seqs -- for instance:

  seq U = 1 2  12  22   122   222    1222    2222 ...
1st diff=  1 10  10  100   100   1000    1000

  seq U'= 10 11 12  22   122   222 ...
1st diff=   1  1  10  100   100

  seq V = 2 4  24  44   244   444    2444    4444 ...
1st diff=  2 20  20  200   200   2000    2000

  seq V'= 20 22 24  44   244   444 ...
1st diff=   2  2  20  200   200

etc.

Questions:
---------
Are there other such seqs using 3 digits but no zero (like 
the seq Q which opens this mail; R was a unfortunate try)?

Could someone compute ALL such 3-digit seq?

What would be the lexicographically first 4-digit seq?

Best,
É.