A349946 -id:A349946 - cp's OEIS frontend

A349947 Triangular array: row n gives the positions of n+1 in A349946.

1, 2, 4, 3, 5, 9, 6, 7, 10, 16, 8, 11, 12, 17, 25, 13, 14, 18, 19, 26, 36, 15, 20, 21, 27, 28, 37, 49, 22, 23, 29, 30, 38, 39, 50, 64, 24, 31, 32, 40, 41, 51, 52, 65, 81, 33, 34, 42, 43, 53, 54, 66, 67, 82, 100, 35, 44, 45, 55, 56, 68, 69, 83, 84, 101, 121

Offset: 1

Clark Kimberling, Dec 07 2021

Every positive integer occurs exactly once, so as a sequence, this is a permutation of the positive integers.
Row n ends in n^2. The first term in row n is (1 + n/1)^2 - 3 if n >= 4 and n is even; as in A028872(n) for n >= 3.
The first term in row n is ((n+1)/2)^2 - 1 if n >= 3 and n is odd, as in A132411(n) for n >= 3.

			First 7 rows:
   1
   2   4
   3   5   9
   6   7  10  16
   8  11  12  17  25
  13  14  18  19  26  36
  14  20  21  27  28  37  49

Cf. A349526, 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}];
q = Map[Total, p]  (* A349946 *)
r = Table[Flatten[Position[q, n]], {n, 2, 12}]  (* A349947 array *)
Flatten[r]  (* A349947 sequence *)

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

Original entry on oeis.org

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

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A349947 Triangular array: row n gives the positions of n+1 in A349946.

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