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A349526 Modified lexicographic ordering of all pairs i,j with 1 <= i <= j; every pair i,j of positive integers occurs exactly once.

%I A349526 #14 Jan 06 2022 16:35:37
%S A349526 1,1,2,2,1,3,2,3,3,1,4,2,4,3,4,4,1,5,2,5,3,5,4,5,5,1,6,2,6,3,6,4,6,5,
%T A349526 6,6,1,7,2,7,3,7,4,7,5,7,6,7,7,1,8,2,8,3,8,4,8,5,8,6,8,7,8,8,1,9,2,9,
%U A349526 3,9,4,9,5,9,6,9,7,9,8,9,9,1,10,2,10,3
%N A349526 Modified lexicographic ordering of all pairs i,j with 1 <= i <= j; every pair i,j of positive integers occurs exactly once.
%C A349526 Concatenate segments: 1 1, then 1 2 2 2, then 1 3 2 3 3 3, etc., so that the general segment is 1 n 2 n ... n n.  This is followed by 1; thus, not only does every i,j with i <= j occur, but so does every i,j with i >= j.  So far, the procedure leaves A349520.  Now, for each number that occurs three times in succession, remove the third occurrence, leaving the present sequence, which has the property that every pair i,j of positive integers occurs exactly once.
%C A349526 The pair n,1 occurs as a(n^2), a(n^2+1).
%C A349526 Is this a duplicate of A329949? - _R. J. Mathar_, Jan 06 2022
%t A349526 t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 10}];
%t A349526 u = Flatten[Partition[t, 2]];
%t A349526 v = Table[n (n + 1), {n, 1, 10}];
%t A349526 Delete[u, Map[{#} &, v]]
%o A349526 (Python)
%o A349526 def auptoj(maxj):
%o A349526     alst = []
%o A349526     for j in range(1, maxj+1):
%o A349526         for i in range(1, j+1):
%o A349526             if i != j: alst.extend([i, j])
%o A349526             else: alst.append(i)
%o A349526     return alst
%o A349526 print(auptoj(10)) # _Michael S. Branicky_, Nov 21 2021
%Y A349526 Cf. A097283, A097285, A339399, A349520, A349946, A349947, A329949.
%K A349526 nonn
%O A349526 1,3
%A A349526 _Clark Kimberling_, Nov 21 2021