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A364057 Lexicographically earliest infinite sequence of positive integers such that every subsequence {a(j), a(j+k), a(j+2k)} (j, k >= 1) is unique.

%I A364057 #28 Nov 04 2023 13:47:14
%S A364057 1,1,1,2,3,1,2,4,5,6,7,8,5,1,2,9,3,4,6,7,1,10,11,12,13,8,14,15,16,3,
%T A364057 17,9,18,4,7,19,5,2,11,12,20,6,1,8,21,22,9,23,24,13,14,3,10,16,17,25,
%U A364057 26,19,27,6,28,11,15,20,22,29,12,21,16,23,30,18,31,32
%N A364057 Lexicographically earliest infinite sequence of positive integers such that every subsequence {a(j), a(j+k), a(j+2k)} (j, k >= 1) is unique.
%C A364057 To find a(n), two criteria must be satisfied:
%C A364057 1. Every subsequence {a(n-2k), a(n-k) a(n)} created by a(n) must be unique.
%C A364057 2. a(n) cannot create the scenario where a future a(m) will create multiple {a(m-2k), a(m-k), a(m)} regardless of choice for a(m). The first time this is the sole reason a candidate is denied is at a(10), see Example below.
%C A364057 Will every subsequence of 3 positive integers appear in arithmetic progression in this sequence?
%C A364057 Will every positive integer occur infinitely many times?
%C A364057 For n >= 3, a(n) != a(n+1).
%C A364057 In the 74 initially published terms, numbers on average seem to reoccur at (very) roughly twice the index of their previous occurrence. This seems worthy of better quantification when further terms are established. - _Peter Munn_, Nov 03 2023
%H A364057 Rémy Sigrist, <a href="/A364057/b364057.txt">Table of n, a(n) for n = 1..10000</a>
%H A364057 Samuel Harkness, <a href="/A364057/a364057.m.txt">MATLAB program</a>
%H A364057 Rémy Sigrist, <a href="/A364057/a364057.txt">C++ program</a>
%e A364057 For a(9), we first try 1. If a(9) were 1, {a(3), a(6), a(9)} would be {1, 1, 1}, but this already occurred at {a(1), a(2), a(3)}.
%e A364057 Next, try 2. If a(9) were 2, {a(3), a(6), a(9)} would be {1, 1, 2}, but this already occurred at {a(2), a(3), a(4)}.
%e A364057 Next, try 3. If a(9) were 3, {a(3), a(6), a(9)} would be {1, 1, 3}, but this already occurred at {a(1), a(3), a(5)}.
%e A364057 Next, try 4. If a(9) were 4, {a(1), a(5), a(9)} would be {1, 3, 4}, but this already occurred at {a(2), a(5), a(8)}.
%e A364057 Then, try 5. New subsequences at indices {a(1), a(5), a(9)} = {1, 3, 5}, {a(3), a(6), a(9)} = {1, 1, 5}, {a(5), a(7), a(9)} = {3, 2, 5}, and {a(7), a(8), a(9)} = {2, 4, 5} are formed, none of which have occurred at any {a(j), a(j+k), a(j+2k)} (for any j and k) previously. No 5 has occurred previously, so criteria (2) in Comments must be satisfied. Thus a(9) = 5.
%e A364057 a(10) is the first time a candidate is denied solely because it would create a guaranteed future duplicate. Note that no subsequences prevent a(10) from being 4.
%e A364057 n    = 1  2  3  4  5  6  7  8  9 10 11 12 13 14
%e A364057 a(n) = 1  1  1  2  3  1  2  4  5 [4]          X
%e A364057                       |           |           |
%e A364057           |                 |                 |
%e A364057 If a(10) were 4, {a(2), a(8), a(14)} = {a(6), a(10), a(14)} = {1, 4, X}, making a subsequence {a(j), a(j+k), a(j+2k)} which is not unique. Therefore a(10) != 4.
%o A364057 (MATLAB) See Links section.
%o A364057 (C++) See Links section.
%Y A364057 Cf. A229037, A362816, A366493, A366624.
%K A364057 nonn
%O A364057 1,4
%A A364057 _Samuel Harkness_, Oct 19 2023