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A366624 Lexicographically earliest sequence of positive integers such that each subsequence enclosed by two equal terms, not including the endpoints, is distinct.

%I A366624 #36 Dec 19 2024 11:45:36
%S A366624 1,1,2,1,2,3,1,2,4,1,2,3,2,1,3,2,1,4,1,2,3,4,1,2,3,5,1,2,3,4,2,1,3,4,
%T A366624 2,1,5,1,2,3,4,5,1,2,3,4,6,1,2,3,4,5,2,1,3,4,5,2,1,4,2,1,3,5,1,2,4,3,
%U A366624 1,2,4,3,2,1,4,3,2,1,5,2,1,3,4,6,1,2,4
%N A366624 Lexicographically earliest sequence of positive integers such that each subsequence enclosed by two equal terms, not including the endpoints, is distinct.
%C A366624 Every positive integer occurs infinitely many times in the sequence.
%C A366624 The subsequence between any two equal terms is unique. For example, consecutive values "A B A" prevents "C B C" because the subsequence "B" would be repeated between equal terms.
%C A366624 Two consecutive values create the empty subsequence, for this reason after {a(1), a(2)} = {1, 1}, no consecutive values will ever occur again.
%C A366624 A new value is always followed by 1.
%H A366624 Michael S. Branicky, <a href="/A366624/b366624.txt">Table of n, a(n) for n = 1..6804</a> (terms 1..1000 from Neal Gersh Tolunsky)
%H A366624 Samuel Harkness, <a href="/A366624/a366624.m.txt">MATLAB program</a>.
%H A366624 Kevin Ryde, <a href="/A366624/a366624.c.txt">C Code</a>.
%e A366624 For a(9), we first try 1. If a(9) were 1, {a(8)} = {2} would be bounded by a(7) = a(9) = 1, but {2} is already bounded by a(2) = a(4) = 1.
%e A366624 Next, try 2. Note this would create consecutive values at {a(8), a(9)}, enclosing the empty subsequence, but this already occurred at {a(1), a(2)}.
%e A366624 Next, try 3. If a(9) were 3, {a(7), a(8)} = {1, 2} would be bounded by a(6) = a(9) = 3, but {1, 2} is already bounded by a(1) = a(4) = 1.
%e A366624 Next, try 4. 4 has yet to appear, so a(9) = 4.
%o A366624 (MATLAB) See Links section.
%o A366624 (Python)
%o A366624 from itertools import islice
%o A366624 def agen(): # generator of terms
%o A366624     m, a = set(), []
%o A366624     while True:
%o A366624         an, allnew = 0, False
%o A366624         while not allnew:
%o A366624             allnew, an, mn = True, an+1, set()
%o A366624             for i in range(len(a)):
%o A366624                 if an == a[i]:
%o A366624                     t = tuple(a[i+1:])
%o A366624                     if t in m or t in mn: allnew = False; break
%o A366624                     mn.add(t)
%o A366624         yield an; a.append(an); m |= mn
%o A366624 print(list(islice(agen(), 87))) # _Michael S. Branicky_, Jan 15 2024
%o A366624 (C) /* See links */
%Y A366624 Cf. A366625 (with distinct multisets), A366631 (with distinct sets), A366493 (including endpoints), A330896, A366691.
%K A366624 nonn
%O A366624 1,3
%A A366624 _Samuel Harkness_, Oct 14 2023