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

%I A366691 #55 Oct 27 2023 10:03:07
%S A366691 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,
%T A366691 2,19,20,21,9,22,5,23,24,10,25,11,26,6,27,28,12,29,30,13,31,14,32,7,
%U A366691 33,15,34,35,36,16,37,17,38,8,39,18,40,41,19,42,43,20
%N A366691 Lexicographically earliest sequence such that each set of terms enclosed by two equal values, excluding the endpoints, contains a distinct number of elements.
%C A366691 The word 'set' means that every element is unique. For example, the set {1,1,2} contains 2 elements (not 3).
%C A366691 Note that we are considering sets between every pair of equal values, not just those that appear consecutively.
%C A366691 Two consecutive values enclose 0 terms, and thus after [a(1), a(2)] = [1, 1], no consecutive equal values occur again.
%H A366691 Neal Gersh Tolunsky, <a href="/A366691/b366691.txt">Table of n, a(n) for n = 1..10000</a>
%H A366691 Rémy Sigrist, <a href="/A366691/a366691.gp.txt">PARI program</a>
%e A366691 a(1)=1; no pair of terms exists yet.
%e A366691 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.
%e A366691 a(3)=2 because if a(3)=1, this would create a second pair enclosing 0 elements.
%e A366691 a(4)=1 creates two new sets: [1, 2, 1], enclosing 1 element {2}, and [1, 1, 2, 1], enclosing 2 elements {1, 2}.
%e A366691 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.
%o A366691 (PARI) See Links section.
%o A366691 (Python)
%o A366691 from itertools import islice
%o A366691 def agen(): # generator of terms
%o A366691     e, a = set(), []
%o A366691     while True:
%o A366691         an, allnew = 0, False
%o A366691         while not allnew:
%o A366691             allnew, an, ndset = True, an+1, set()
%o A366691             for i in range(len(a)):
%o A366691                 if an == a[i]:
%o A366691                     nd = len(set(a[i+1:]))
%o A366691                     if nd in e or nd in ndset: allnew = False; break
%o A366691                     ndset.add(nd)
%o A366691         yield an; a.append(an); e |= ndset
%o A366691 print(list(islice(agen(), 72))) # _Michael S. Branicky_, Oct 25 2023
%Y A366691 Cf. A337226 (with nondistinct terms counted), A330896, A363757, A366631.
%K A366691 nonn
%O A366691 1,3
%A A366691 _Neal Gersh Tolunsky_, Oct 17 2023
%E A366691 More terms from _Rémy Sigrist_, Oct 25 2023