ACRONYMS : A Common Representation Of Names You Must Shorten

[Eric Angelini]

Hello SeqFans & Math-Fun,
If those two seq. are of interest, could someone help in providing more terms?
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First digit of a(n) is given by the succession of digits in S
[a(n+1) is the smallest available integer not yet present in S]:
S= 0,1,2,3,4,5,6,7,8,9,11,12,13,21,14,31,22,15,16,41,32,17,23,24,17,51,18,61...
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First digit of a(n) is given by the succession of digits in S
[S is monotonicaly increasing]:
S= 0,1,2,3,4,5,6,7,8,9,11,12,13,21,111,311,2111,11111,11112,11113,11114,31111,...
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Best,
É.