A349947 -id:A349947 - cp's OEIS frontend

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

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, 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, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1, 10, 2, 10, 3

Offset: 1

Clark Kimberling, Nov 21 2021

nonn

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.
The pair n,1 occurs as a(n^2), a(n^2+1).
Is this a duplicate of A329949? - R. J. Mathar, Jan 06 2022

Cf. A097283, A097285, A339399, A349520, A349946, A349947, A329949.

t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 10}];
u = Flatten[Partition[t, 2]];
v = Table[n (n + 1), {n, 1, 10}];
Delete[u, Map[{#} &, v]]

def auptoj(maxj):
    alst = []
    for j in range(1, maxj+1):
        for i in range(1, j+1):
            if i != j: alst.extend([i, j])
            else: alst.append(i)
    return alst
print(auptoj(10)) # Michael S. Branicky, Nov 21 2021

A349946 a(n) = A349526(n) + A349526(n+1).

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 5, 6, 4, 5, 6, 6, 7, 7, 8, 5, 6, 7, 7, 8, 8, 9, 9, 10, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15

Offset: 1

Clark Kimberling, Dec 07 2021

nonn

Every positive integer n >= 2 occurs exactly n-1 times; the last occurrence of n is a((n-1)^2).

			A349426 = (1,1,2,2,1,3,2,3,3,1,4,2,4,3,4,4,1,...), in which every pair i,j of positive integers occurs exactly once; a(1) = 1+1, a(2) = 1+2, a(3) = 2+2.

Cf. A349526, A349947 (array: row n shows positions of n+1 in A349946).

t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 100}];
u = Flatten[Partition[t, 2]];
v = Table[n (n + 1), {n, 1, 80}];
d = Delete[u, Map[{#} &, v]]; (* A349526 *)
p = Table[{d[[n]], d[[n + 1]]}, {n, 1, 150}];
Map[Total, p]  (* A349946 *)

cp's OEIS Frontend

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

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