[seqfan] Distinct strings

[Éric Angelini]

Hello SeqFans,
Could someone check and compute perhaps a few more terms
(if this is of interest)?
S = 1, 2, 222, 22, 2222, 3, 120, 10, 234, 24, 103, 112, 122, 100, 2345, 25, 1130, 102, 1345, 111, 2340, 1000,...

We concatenate a(1) and a(2) to form 12.
12 produces 3 distinct strings: (12,1,2)
We concatenate a(2) and a(3) to form 2222.
2222 produces 4 distinct strings: (222,222,22,2)
We concatenate a(3) and a(4) to form 22222.
22222 produces 5 distinct strings: (22222,2222,222,22,2)
We concatenate a(4) and a(5) to form 222222.
222222 produces 6 distinct strings: (222222,22222,2222,222,22,2)
We concatenate a(5) and a(6) to form 22223.
22223 produces 9 distinct strings: (22223,2222,2223,222,223,22,23,2,3)
etc.

The rules:
Lexico-first seq of distinct terms;
Extend S with the smallest integer such that the new concatenation
beats the previous one by the smallest possible margin (in terms of
strings quantity).

This might be not clear and I will explain with an example:

Say we start T with:

T = 1, 2, ...
We could extend T with 10 as 210 produces 6 strings (210,21,10,2,1,0)
-- and 6 strings "beats" the 3 strings of 12;
What about extending T with 11 istead of 10? We have now 211 which
produces 5 strings (211,21,11,2,1) which still beats the 3 strings of 12,
but by a smaller margin (2 instead of 3);
The best result comes from a(3) = 222 (and not 10 or 11) as 2222 
produces 4 strings (beats 3).

BTW, 0 is a valid string, but not 01 (2019 would produce 
2019,201,20,19,2,0,1,9 and not 019 or 01).

I'm afraid the computing time will increase dramatically with the next terms
(said my friend Solal Streeter who fought with his machine all night long !-)
Best,
É.