A210415 List the positions of all digits 1 in the concatenation of all terms, not necessarily in order. This is the lexicographically earliest such sequence.

1, 3, 10, 6, 11, 7, 21, 13, 15, 17, 19, 101, 24, 100, 29, 102, 34, 103, 39, 104, 44, 105, 49, 106, 54, 107, 59, 108, 64, 109, 69, 110, 70, 76, 111, 77, 78, 85, 112, 86, 91, 94, 113, 95, 211, 1111, 11111, 1110, 115, 116, 118, 119, 121, 122, 124, 125, 127, 129

Offset

1,2

Comments

Original name: "A self-describing sequence: The a(n) say the positions of the digits 1 inside the sequence when it is read as a string of digits."

For each n, the digit in position a(n) is equal to 1. At each step, choose the minimum integer not yet present in the sequence and not leading to a contradiction.

This sequence had been mentioned by Wasserman in 2008, cf. A098645, a variant of this sequence with additional restriction a(n+1) > a(n). - M. F. Hasler, Oct 08 2013

Links

Danny Rorabaugh, Table of n, a(n) for n = 1..1000

Index to the OEIS: Entries related to self-referencing sequences.

Example

The sequence starts with 1: the first digit is equal to 1. In the second position we cannot write 2 because the second digit would not be 1 but 2. Then we write 3. The third digit must be 1 and the minimum number starting with 1 is 10. And so on.

Prog

(Sage) #Returns the first n terms of this sequence for digit d
def dig_loc(d, n):
  L, S = [], ""
  while len(L)<n:
    ext, new, lenS = 0, 0, len(S)
    while new==0:
      ext += 1
      while d==0 and (lenS+ext+1 in L): ext += 1
      d_s = [i for i in range(ext) if (lenS+i+1 in L)]
      need = ext - len(d_s)
      for i in range(10^(need-1)*(0 not in d_s), 10^(need)):
        a = [str(0)]*(need - len(str(i))) + list(str(i))
        for j in d_s: a.insert(j, d)
        new = sum([int(a[ext-1-j])*10^j for j in range(ext)])
        if ( new not in L ) and ( new>lenS or S[new-1]==str(d) ) and ( (new-lenS-1 not in range(ext)) or a[new-lenS-1]==str(d) ) and ( d!=0 or lenS+ext+1!=new ):
          L.append(new)
          S += str(new)
          break
        else: new = 0
  return L
dig_loc(1, 58) # Danny Rorabaugh, Nov 27 2015

Crossrefs

This construction for other decimal digits is A210414, A210416, A210417, A210418, A210419, A210420, A210421, A210422, A210423.

Variants of this sequence include A098645, A098670, A114134, A167519.

Sequence in context: A195919 A275741 A244846 * A337275 A087397 A210414

Adjacent sequences: A210412 A210413 A210414 * A210416 A210417 A210418

Keyword

nonn,base,nice

Author

Paolo P. Lava, Mar 26 2012

Extensions

Edited by M. F. Hasler, Oct 10 2013

Status

approved