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

%I A378381 #23 Nov 28 2024 11:26:22
%S A378381 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,
%T A378381 19,20,10,21,6,22,23,11,24,12,25,26,13,27,3,28,29,30,31,14,32,15,33,
%U A378381 16,34,8,35,36,17,37,9,38,39,18,40,5,41,42,43,19,44,20,45
%N A378381 Lexicographically earliest sequence such that each set of terms enclosed by two equal values, including the endpoints, contains a distinct number of elements.
%C A378381 The word 'set' means that every element is unique. For example, the set {1,1,2} contains 2 elements (not 3).
%C A378381 Note that we are considering sets between every pair of equal values, not just those that appear consecutively.
%C A378381 Two consecutive values enclose 1 term, and thus after [a(1), a(2)] = [1, 1], no consecutive equal values occur again.
%H A378381 Neal Gersh Tolunsky, <a href="/A378381/b378381.txt">Table of n, a(n) for n = 1..10000</a>
%e A378381 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.
%o A378381 (Python)
%o A378381 from itertools import islice
%o A378381 def agen(): # generator of terms
%o A378381     e, a = set(), []
%o A378381     while True:
%o A378381         an, allnew = 0, False
%o A378381         while not allnew:
%o A378381             allnew, an, ndset = True, an+1, set()
%o A378381             for i in range(len(a)):
%o A378381                 if an == a[i]:
%o A378381                     nd = len(set(a[i:]))
%o A378381                     if nd in e or nd in ndset: allnew = False; break
%o A378381                     ndset.add(nd)
%o A378381         yield an; a.append(an); e |= ndset
%o A378381 print(list(islice(agen(), 73))) # _Michael S. Branicky_, Nov 26 2024
%Y A378381 Cf. A366691.
%K A378381 nonn
%O A378381 1,3
%A A378381 _Neal Gersh Tolunsky_, Nov 26 2024