A378381 Lexicographically earliest sequence such that each set of terms enclosed by two equal values, including the endpoints, contains a distinct number of elements.

1, 1, 2, 3, 2, 4, 3, 5, 6, 4, 7, 3, 8, 9, 5, 10, 6, 11, 12, 7, 13, 3, 14, 15, 16, 8, 17, 9, 18, 5, 19, 20, 10, 21, 6, 22, 23, 11, 24, 12, 25, 26, 13, 27, 3, 28, 29, 30, 31, 14, 32, 15, 33, 16, 34, 8, 35, 36, 17, 37, 9, 38, 39, 18, 40, 5, 41, 42, 43, 19, 44, 20, 45

Offset: 1

Author

Neal Gersh Tolunsky, Nov 26 2024

Keywords

• nonn

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 1 term, and thus after [a(1), a(2)] = [1, 1], no consecutive equal values occur again.

Examples

			a(4) cannot be 1 since this would create a second pair enclosing two values, [1,2,1] being an equivalent set to [1,2,1,1]. We cannot have a(4)=2 because [1,2,1] would enclose the same number of elements as [2,1,2]. So a(4)=3, which has not occurred before.
		

Links

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

Crossrefs

Cf. A366691.

Programs

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:]))
                    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(), 73))) # Michael S. Branicky, Nov 26 2024