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A370264 Lexicographically earliest sequence such that each subsequence enclosed by a pair of equal values, including the endpoints, has a unique sum.

%I A370264 #11 Feb 17 2024 12:41:39
%S A370264 1,1,2,1,3,2,1,3,3,4,2,1,3,5,4,2,6,7,1,3,5,4,7,6,2,8,1,5,6,9,9,3,1,10,
%T A370264 2,8,4,1,10,6,9,3,2,5,11,12,4,3,10,7,8,2,13,11,12,4,13,1,14,3,9,15,5,
%U A370264 6,7,14,16,6,2,4,8,12,3,9,10,11,5,7,13,1,14
%N A370264 Lexicographically earliest sequence such that each subsequence enclosed by a pair of equal values, including the endpoints, has a unique sum.
%C A370264 Note that we are considering the sums of the terms between every pair of equal values, not just those that appear consecutively.
%H A370264 Michael S. Branicky, <a href="/A370264/b370264.txt">Table of n, a(n) for n = 1..10000</a>
%e A370264 a(2)=1 creates the pair [a(1), a(2)] = [1, 1], which gives the unique sum of 2.
%e A370264 a(4)=1 creates two unique sums: [1,2,1] = sum of 4 and [1,1,2,1] = sum of 5.
%e A370264 a(8)=3 creates one unique sum: [3,2,1,3] = sum of 9.
%o A370264 (Python)
%o A370264 from itertools import islice
%o A370264 def agen(): # generator of terms
%o A370264     s, a = set(), []
%o A370264     while True:
%o A370264         an, allnew = 0, False
%o A370264         while not allnew:
%o A370264             allnew, an, sn = True, an+1, set()
%o A370264             for i in range(len(a)):
%o A370264                 if an == a[i]:
%o A370264                     t = sum(a[i+1:]) + 2*an
%o A370264                     if t in s or t in sn: allnew = False; break
%o A370264                     sn.add(t)
%o A370264         yield an; a.append(an); s |= sn
%o A370264 print(list(islice(agen(), 81))) # _Michael S. Branicky_, Feb 14 2024
%Y A370264 Cf. A370264 (excluding endpoints), A366493, A366624, A366631, A366625.
%K A370264 nonn
%O A370264 1,3
%A A370264 _Neal Gersh Tolunsky_, Feb 13 2024
%E A370264 a(16) and beyond from _Michael S. Branicky_, Feb 14 2024