True So Far

[Eric Angelini]

Hello math-fun and seqfan,

I've just sent this to the OEIS :

    10 12 13 14 15 16 17 18 19 20 23 24 25 26 27 28 29 30
    34 35 36 37 38 39 40 45 46 47 48 49 50 56 57 58 59 60
    67 68 69 70 78 79 80 89 90 90 102 103 104 105 106 107
    108 109 112 113 114 115 116 117 118 119 123 124 125
    126 127 128 129 134 135 136 137 138 139 145 146 147
    148 149 156 157 158 159 167 168 169 178 179 180 189...

[more hand calculated terms here (hope no errors)]:

http://www.cetteadressecomportecinquantesignes.com/TrueSoFar.htm

Description :

    "True so far"-sequence. Last digit of a(n) must be seen
    as a glyph and preceding digits as a quantity. So "10"
    reads [one "0"] and "12" [one "2"] -- which are both true
    statements: there is only one "0" glyph so far in the
    sequence when [10] is read, and there is only one "2"
    glyph when [12] is read. The sequence is built with
    [a(n+1)-a(n)] being minimal and a(n+1) always "true so
    far". This explains why integers [11], [21], [22], [31],
    etc. are not in: their statements are false.

    The nice substring ...1112,1113,1114,1115,1116,1117 1118...
    appears in the sequence -- which means that so far the
    whole sequence has used 111 "2", 111 "3", 111 "4", 111 "5",
    111 "6", 111 "7" and 111 "8"...

Question which ruined my sleep tonight:

« Will the sequence ever stop? »

... my intuition says yes...
... could someone compute this and check for some more integers?

Thanks,
É.
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