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A102357
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"True so far" sequence: floor(a(n)/10) is the number of digits (a(n) mod 10) within the first n terms; a(n) is the smallest such number larger than a(n-1).
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14
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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, 102, 103, 104, 105, 106, 107, 108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 123
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OFFSET
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1,1
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COMMENTS
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Original definition (edited): In a(n), the last digit must be seen as a glyph and preceding digits as a number, counting occurrences of the glyph up to and including a(n). "10" reads [one '0'] and "12" [one '2'] - which are both true statements: there is one '0' glyph so far in the sequence when 10 is read and there is one '2' glyph when 12 is read. The sequence is built with a(n+1)-a(n) being minimal, positive, and a(n) always "true so far". This explains why there are no integers 11, 21, 22, 31 etc.: their statements are false.
The substring ...1112,1113,1114,1115,1116,1117... appears in the sequence - which means that so far the whole sequence has used 111 '2's, 111 '3's, 111 '4's, 111 '5's, 111 '6's and 111 '7's. [Corrected (1118 is not in the sequence!) by M. F. Hasler, Nov 18 2019]
The sequence is finite. The last term is a(2024) = 8945. The largest terms ending with each digit appear to be: 5890, 8201, 8312, 8623, 8734, 8945, 7756, 6697, 6778, 5979. - Chuck Seggelin, Feb 22, 2005 [Corrected '8495' but other terms unverified. - M. F. Hasler, Nov 18 2019]
When this sequence terminates there are 624 zero, 822 ones, 834 twos, 864 threes, 874 fours, 894 fives, 779 sixes, 697 sevens, 697 eights and 617 nines. - Robert G. Wilson v, Feb 22(?) 2005
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LINKS
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MATHEMATICA
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a[0] = {}; a[n_] := a[n] = Block[{k = Max[a[n - 1], 0], b = Sort[ Flatten[ Table[ IntegerDigits[ a[i]], {i, 0, n - 1}] ]]}, While[ Count[ Join[b, IntegerDigits[ IntegerPart[k/10]]], Mod[k, 10]] != IntegerPart[k/10], k++ ]; k]; Table[ a[n], {n, 63}] (* Robert G. Wilson v, Feb 22 2005 *)
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PROG
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(PARI) c=Vec(0, 10); a=10; for(n=1, 2024, while(a\10<=c[a%10+1] || a\10 != c[a%10+1]+#select(d->d==a%10, digits(a)), a++); [c[d+1]++|d<-digits(a)]; print1(a", ")) \\ M. F. Hasler, Nov 18 2019
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CROSSREFS
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KEYWORD
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base,easy,nonn,fini,full
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AUTHOR
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EXTENSIONS
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Edited and shorter definition from M. F. Hasler, Nov 18 2019
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STATUS
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approved
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