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

Author

Neal Gersh Tolunsky, May 10 2024

Keywords

• nonn

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.

Examples

			  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
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Neal Gersh Tolunsky, Graph of first differences of first 20000 terms.

Crossrefs

Cf. A364603, A364604.

Programs

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

Extensions

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