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A378821 Lexicographically earliest sequence of distinct positive integers such that the count of integers between a(n) and a(n-1), excluding values already in the sequence, is distinct from the same count for any other a(k) and a(k-1) at the time they occurred.

%I A378821 #23 Dec 21 2024 23:52:08
%S A378821 1,2,4,7,11,3,10,18,5,16,26,6,22,35,8,27,42,9,32,51,12,37,58,13,41,66,
%T A378821 14,47,74,15,53,83,17,57,90,19,62,98,20,68,106,21,73,115,23,78,122,24,
%U A378821 84,131,25,88,138,28,94,147,29,99,154,30,103,162,31,109,170,33
%N A378821 Lexicographically earliest sequence of distinct positive integers such that the count of integers between a(n) and a(n-1), excluding values already in the sequence, is distinct from the same count for any other a(k) and a(k-1) at the time they occurred.
%H A378821 Michael S. Branicky, <a href="/A378821/b378821.txt">Table of n, a(n) for n = 1..10000</a>
%e A378821  a(1-5) = 1,2,4,7,11 follow a straightforward pattern of counting up, where each pair of consecutive terms encloses (in order) 0,1,2,3 unused values.
%e A378821   1,2,3,4,5,6,7,8,9,10,11
%e A378821   1 2 * 4 * * 7 * * *  11
%e A378821   ^ ^   ^     ^        ^
%e A378821   At a(6), a(5)=11 and a(6)=3 enclose 5 unused values:
%e A378821   1,2,3,4,5,6,7,8,9,10,11
%e A378821   1 2 3 4 * * 7 * * *  11
%e A378821       ^                ^
%o A378821 (Python)
%o A378821 from itertools import count, islice
%o A378821 def c(k, m, a): return sum(1 for i in range(min(k, m)+1, max(k, m)) if i not in a)
%o A378821 def agen(): # generator of terms
%o A378821     a, d, an, m = set(), set(), 1, 2
%o A378821     while True:
%o A378821         yield an
%o A378821         a.add(an)
%o A378821         found, k = False, m
%o A378821         if m < an:
%o A378821             ck = c(k, an, a)
%o A378821             for k in range(m, an):
%o A378821                 if k not in a:
%o A378821                     if ck not in d:
%o A378821                         found = True
%o A378821                         break
%o A378821                     ck -= 1
%o A378821         if not found:
%o A378821             kk = max(m, an+1)
%o A378821             ck = c(kk, an, a)
%o A378821             for k in count(kk):
%o A378821                 if k not in a:
%o A378821                     if ck not in d:
%o A378821                         found = True
%o A378821                         break
%o A378821                     ck += 1
%o A378821         d.add(c(an, k, a))
%o A378821         an = k
%o A378821         while m in a: m += 1
%o A378821 print(list(islice(agen(), 66))) # _Michael S. Branicky_, Dec 09 2024
%Y A378821 Cf. A378822, A081145, A091263.
%K A378821 nonn
%O A378821 1,2
%A A378821 _Neal Gersh Tolunsky_, Dec 08 2024