A097283 -id:A097283 - cp's OEIS frontend

A097284 Rectangular array T by antidiagonals: T(n,k) = rank of k-th n in A097283.

1, 3, 2, 5, 7, 4, 9, 11, 8, 6, 13, 15, 17, 12, 10, 19, 21, 23, 18, 16, 14, 25, 27, 29, 31, 24, 22, 20, 33, 35, 37, 39, 32, 30, 28, 26, 41, 43, 45, 47, 49, 40, 38, 36, 34, 51, 53, 55, 57, 59, 50, 48, 46, 44, 42, 61, 63, 65, 67, 69, 71, 60, 58, 56, 54, 52, 73, 75, 77, 79, 81, 83

Offset: 1

Clark Kimberling, Aug 05 2004

As a sequence, this is a permutation of the natural numbers.

			Northwest corner:
1 3 5 9 13
2 7 11 15 21
4 8 17 23 29
6 12 18 31 39

Cf. A097283.

A349520 Let S_k denote the list of pairs (1,k), (2,k), (3,k), ..., (k,k); sequence lists the pairs in S_1, S_2, S_3, ...

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 20 2021

nonn

Concatenate segments: 1 1, then 1 2 2 2, then 1 3 2 3 3 3, 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. Every pair i,j of positive integers with i < j or i > j occurs exactly once.

David A. Corneth, Table of n, a(n) for n = 1..10100

Cf. A084855, A097283, A097285, A339399.

t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 10}];
Flatten[Partition[t, 2]]

def auptoj(maxj):
    alst = []
    for j in range(1, maxj+1):
        for i in range(1, j+1):
            alst.extend([i, j])
    return alst
print(auptoj(9)) # Michael S. Branicky, Nov 21 2021

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

A097289 Contains exactly once every pair (i,j) satisfying 0 < i <= j.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 1, 4, 4, 2, 3, 1, 5, 5, 2, 4, 1, 6, 6, 2, 5, 3, 4, 1, 7, 7, 2, 6, 3, 5, 1, 8, 8, 2, 7, 3, 6, 4, 5, 1, 9, 9, 2, 8, 3, 7, 4, 6, 1, 10, 10, 2, 9, 3, 8, 4, 7, 5, 6, 1, 11, 11, 2, 10, 3, 9, 4, 8, 5, 7, 1, 12, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 1, 13, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 1

Offset: 1

Clark Kimberling, Aug 05 2004

nonn

All pairs (i,j) having i>j occur (not necessarily uniquely) except those of the form (i,i-1) for i>=3. (Those are included at A097291.)

Cf. A097290, A097291.

Obtained from A097283 by inserting m right after the first occurrence of m, for each positive integer m.

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A097284 Rectangular array T by antidiagonals: T(n,k) = rank of k-th n in A097283.

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A349520 Let S_k denote the list of pairs (1,k), (2,k), (3,k), ..., (k,k); sequence lists the pairs in S_1, S_2, S_3, ...

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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.

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Python

A097289 Contains exactly once every pair (i,j) satisfying 0 < i <= j.

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