A102357 "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).

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

Offset: 1

Author

Eric Angelini, Feb 21 2005

Keywords

• base
• easy
• nonn
• fini
• full

Comments

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.

Terms must increase. Without this condition we obtain A102850. - David Wasserman, Feb 13 2008

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

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..2024 (based on C. Seggelin's data)
• Eric Angelini, Sequence True-so-far
• Eric Angelini, Sequence True-so-far [Cached copy with permission]
• C. Seggelin, Sequence True-So-Far

Crossrefs

Cf. A102850.

Programs

Mathematica

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 *)

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

Extensions

Chuck Seggelin and David W. Wilson both computed the full 2024 terms

Offset corrected by Nathaniel Johnston, May 17 2011

Edited and shorter definition from M. F. Hasler, Nov 18 2019