A366691 Lexicographically earliest sequence such that each set of terms enclosed by two equal values, excluding the endpoints, contains a distinct number of elements.

1, 1, 2, 1, 3, 4, 2, 5, 6, 3, 7, 4, 8, 2, 9, 5, 10, 11, 6, 12, 3, 13, 14, 7, 15, 4, 16, 17, 8, 18, 2, 19, 20, 21, 9, 22, 5, 23, 24, 10, 25, 11, 26, 6, 27, 28, 12, 29, 30, 13, 31, 14, 32, 7, 33, 15, 34, 35, 36, 16, 37, 17, 38, 8, 39, 18, 40, 41, 19, 42, 43, 20

Offset

1,3

Comments

The word 'set' means that every element is unique. For example, the set {1,1,2} contains 2 elements (not 3).

Note that we are considering sets between every pair of equal values, not just those that appear consecutively.

Two consecutive values enclose 0 terms, and thus after [a(1), a(2)] = [1, 1], no consecutive equal values occur again.

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program

Example

a(1)=1; no pair of terms exists yet.
a(2)=1 creates the pair [1, 1], which encloses 0 elements. This means that no consecutive equal values can occur again, since this would create another set of 0 elements.
a(3)=2 because if a(3)=1, this would create a second pair enclosing 0 elements.
a(4)=1 creates two new sets: [1, 2, 1], enclosing 1 element {2}, and [1, 1, 2, 1], enclosing 2 elements {1, 2}.
a(5) cannot be 1 as this would again create a pair enclosing 0 elements [1,1]. 2 would create the pair [2, 1, 2] which encloses 1 element {1}, which has been impossible since a(4). So a(5)=3, which has not occurred before.

Prog

(PARI) See Links section.
(Python)
from itertools import islice
def agen(): # generator of terms
    e, a = set(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, ndset = True, an+1, set()
            for i in range(len(a)):
                if an == a[i]:
                    nd = len(set(a[i+1:]))
                    if nd in e or nd in ndset: allnew = False; break
                    ndset.add(nd)
        yield an; a.append(an); e |= ndset
print(list(islice(agen(), 72))) # Michael S. Branicky, Oct 25 2023

Crossrefs

Cf. A337226 (with nondistinct terms counted), A330896, A363757, A366631.

Sequence in context: A358799 A125160 A359027 * A009947 A166711 A026249

Adjacent sequences: A366688 A366689 A366690 * A366692 A366693 A366694

Keyword

nonn

Author

Neal Gersh Tolunsky, Oct 17 2023

Extensions

More terms from Rémy Sigrist, Oct 25 2023

Status

approved