Seq. of terms = 1, 10, 100, 101, 2012, 10121?...: T = 1'10'100'1 01'2 012'10121
Common substr. = 1, 10, 10, 01, 012, ... : S = 1'10'10'01'01 2'...
(=> concatenation of the next L.C.S. must give "012'10121'...")
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After a(1) = 1, the next larger number sharing a digit with 1 is a(2) = 10.
The concatenated terms so far give T = "110(...)", and the (concatenated) common substring(s) give S = "1(...)". Since S must equal T, the concatenation of the next longest common substrings (LCS) must yield "10(...)".
So the next term a(3) must share a digit 1 with a(2), but it must also contain a digit 0 to produce the digit 0 subsequently required in S.
The next larger number which satisfies both of these requirements is a(3) = 100. This gives S = "110100(...)" and T = "110(...)". Concatenation of subsequent longest common substrings (LCS) must therefore give "100(...)".
As before it is impossible to have as LCS only one digit "1", because then there may be no '0' in the next term, but the next digit in T must be a '0'.
So we must have LCS(a(3), a(4)) = "10" (or longer, but this would require a(4) >= 1000 and we will find a smaller solution).
The next larger possible term, a(4) = 101, indeed satisfies the constraints, and we will see that it does not lead to contradictions.
This gives T = "1'10'100'101(...)" and S = "1'10'10(...)", so concatenation of subsequent LCS must produce "01'01(...)". (We use a separator ' for better readability, but this is not to be considered an element of the string.)
As before it is impossible to have the next LCS equal to "0", because then there may be no '1' in the next term, but the subsequent digit in S must be a '1'.
So we look for a(5) with LCS(a(4), a(5)) = "01" (but no "10" in a(5) as to have a welldefined LCS). The subsequent LCS(a(5),a(6)) and following must then produce "01"+a(5)+"(...)".
We find that 201 is not possible for a(5): This would require S to go on with "012(...)", so we'd again need LCS(a(5),a(6)) = "01" but no "20" in a(6), but this allows no acceptable a(7) such that LCS(a(6),a(7)) starts with the required '2'. Similarly, 301, ..., 901 are not possible.
Also the next larger choice (with welldefined LCS) 1201 is not possible. (We'd need a(6) with "01", but no "12" nor "20", but then a(7) would need an LCS "12" with a(6): impossible.)
The next larger possible choice is a(5) = 201x for some digit x > 0. We find that there can't be a solution with LCS(a(5), a(6)) of length <= 2, but it is possible for LCS = "012", whence x = 2 finally yields a solution. It gives T = "1'10'100'101'2012(...)" and S = "1'10'10'01(...)", so the next LCS must give "012012(...)", and for example a(6) = "10121" appears to give a valid solution.
