A372704 a(1)=1; a(2)=2; for n>1, this is the lexicographically earliest sequence such that, following each occurrence of a(n), a(n) is banned for the next k terms, where k is the number of terms prior to a(n) that are not equal to a(n).

1, 2, 1, 2, 1, 3, 2, 1, 4, 5, 6, 2, 1, 3, 7, 8, 9, 4, 10, 5, 2, 1, 6, 11, 12, 13, 3, 14, 15, 7, 16, 8, 17, 9, 4, 18, 19, 2, 1, 5, 10, 20, 21, 22, 6, 23, 24, 11, 25, 12, 26, 3, 13, 27, 28, 14, 29, 15, 7, 30, 31, 16, 8, 32, 33, 17, 9, 4, 34, 35, 2, 1, 18, 19, 36

Offset

1,2

Comments

1 occurs immediately after its ban ends so that the i-th occurrence of a(n) = 1, for i >= 2, is at n = 2^(i-2) + i = A052968(i-1).

2 occurs immediately after its ban ends, since it turns out that's immediately before the ban on 1 ends, so the i-th occurrence of a(n) = 2 is at n = 2^(i-1) + i = A005126(i-1).

If the definition is changed so that k is the number of terms in the sequence thus far equal to a(n) (rather than unequal), this becomes A002260 without the initial 1.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Neal Gersh Tolunsky, Graph of first differences of first 20000 terms.

Example

  a(n)   ban 1  2  3  4  5  6 ...
   1         |  |  |  |  |  |
   2         |  |  |  |  |  |
   1         |  x  |  |  |  |
   2         x  |  |  |  |  |
   1         |  x  |  |  |  |
   3         x  x  |  |  |  |
   2         x  |  x  |  |  |
   1         |  x  x  |  |  |
   4         x  x  x  |  |  |
   5         x  x  x  x  |  |
   6         x  x  x  x  x  |
   2         x  |  |  x  x  x
   1         |  x  |  x  x  x
   3         x  x  x  x  x  x

Prog

(Python)
from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    an, ban, occurs = 1, {1: 1}, Counter([1])
    for n in count(2):
        yield an
        an = next(k for k in count(1) if k not in ban)
        for k in list(ban):
            if ban[k] > 1: ban[k] -= 1
            else: del ban[k]
        ban[an] = n - 1 - occurs[an]
        occurs[an] += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, May 10 2024

Crossrefs

Cf. A364603, A364604.

Sequence in context: A205696 A029635 A375577 * A203301 A309853 A107456

Adjacent sequences: A372701 A372702 A372703 * A372705 A372706 A372707

Keyword

nonn

Author

Neal Gersh Tolunsky, May 10 2024

Extensions

a(20) and beyond from Michael S. Branicky, May 10 2024

Status

approved