A097285 -id:A097285 - cp's OEIS frontend

A097286 Rectangular array T by descending antidiagonals: T(n,k) = rank of k-th n in A097285.

1, 3, 2, 7, 5, 4, 13, 9, 6, 8, 21, 15, 11, 10, 14, 31, 23, 17, 12, 16, 22, 43, 33, 25, 19, 18, 24, 32, 57, 45, 35, 27, 20, 26, 34, 44, 73, 59, 47, 37, 29, 28, 36, 46, 58, 91, 75, 61, 49, 39, 30, 38, 48, 60, 74, 111, 93, 77, 63, 51, 41, 40, 50, 62, 76, 92, 133, 113, 95, 79, 65, 53, 42, 52, 64, 78, 94, 112

Offset: 1

Clark Kimberling, Aug 05 2004

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

			Corner:
   1    3    7   13    21    31    43    57    73    91   111
   2    5    9   15    23    33    45    59    75    93   113
   4    6   11   17    25    35    47    61    77    95   115
   8   10   12   19    27    37    49    63    79    97   117
  14   16   18   20    29    39    51    65    81    99   119
  22   24   26   28    30    41    53    67    83   101   121
  32   34   36   38    40    42    55    69    85   103   123
  44   46   48   50    52    54    56    71    87   105   125
  58   60   62   64    66    68    70    72    89   107   127
  74   76   78   80    82    84    86    88    90   109   129
  92   94   96   98   100   102   104   106   108   110   131

Cf. A002061 (row 1), A014206 (column 1), A097285.

s = {1, 2}; Do[s = Join[s, Riffle[Range[n - 1], n], {n}], {n, 3, 12}];
Grid[Table[Flatten[Position[s, n]], {n, 1, 12}]]  (* Clark Kimberling, May 09 2025 *)

A097287 Rectangular array T by antidiagonals: T(n,k) = rank of n where (n,k) occurs in A097285, for distinct n,k and T(n,n)=0 for all n.

Original entry on oeis.org

0, 1, 2, 3, 0, 6, 7, 5, 4, 12, 13, 9, 0, 8, 20, 21, 15, 11, 10, 14, 30, 31, 23, 17, 0, 16, 22, 42, 43, 33, 25, 19, 18, 24, 32, 56, 57, 45, 35, 27, 0, 26, 34, 44, 72, 73, 59, 47, 37, 29, 28, 36, 46, 58, 90, 91, 75, 61, 49, 39, 0, 38, 48, 60, 74, 110, 111, 93, 77, 63, 51, 41, 40, 50, 62

Offset: 1

Clark Kimberling, Aug 05 2004

Every odd number occurs exactly once above the diagonal and every even, exactly once below the diagonal.

			Northwest corner:
0 1 3 7 13
2 0 5 9 15
6 4 0 11 17
12 8 10 0 19
T(1,4)=7 because in A097285 the unique occurrence of 1,4 is
in positions 7,8.

A349550 Meta-Wythoff array based on A097285: M = (M(n,k)), by downward antidiagonals; every row of M is eventually a row of the Wythoff array, W = A035513, and every row of W is a row of M; see Comments.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 5, 4, 3, 1, 8, 7, 5, 4, 2, 13, 11, 8, 5, 4, 3, 21, 18, 13, 9, 6, 4, 1, 34, 29, 21, 14, 10, 7, 5, 2, 55, 47, 34, 23, 16, 11, 6, 5, 3, 89, 76, 55, 37, 26, 18, 11, 7, 5, 4, 144, 123, 89, 60, 42, 29, 17, 12, 8, 5, 1, 233, 199, 144, 97, 68, 47

Offset: 1

Clark Kimberling, Nov 21 2021

Suppose that (s(1), s(2), ...) is a sequence satisfying s(k) = s(k-1) + s(k-2) for k >= 3. If s(1) and s(2) are positive integers, then there is an index n such that (s(n), s(n+1), ...) is a row of A035513. The n-th row of M is the sequence (s(1), s(2), ...), where (s(1), s(2)) are the n-th pair described in A097285.

Every row of W is a row of M; indeed, M consists of all tails of all rows of W.

			Corner:
  1, 2, 3,  5,  8, 13, 21, 34,  55,  89, 144, 233
  1, 3, 4,  7, 11, 18, 29, 47,  76, 123, 199, 322
  2, 3, 5,  8, 13, 21, 34, 55,  89, 144, 233, 377
  1, 4, 5,  9, 14, 23, 37, 60,  97, 157, 254, 411
  2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466
  3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521
  1, 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500
  2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555
  3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
  4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665
Example: The first 7 pairs in A097285 are (1,2), (1,3), (2,3), (1,4), (2,4), (3,4), (1,5), so that the first 7 rows of M are
(1,2,3,5,8,...) = (row 1 of W) = Fibonacci numbers, A000045;
(1,3 4,7,11,...), which includes row 2 of W, the Lucas numbers, A000032;
(2,3,5,8,13,...), a tail of row 1 of W;
(1,4,5,9,14,...), which includes row 4 of W;
(2,4,6,10,16,...), which includes row 3 of W;
(3,4,7,11,18,...), which includes row 2 of W;
(1,5,6,11,17,...), which includes row 7 of W.

Cf. A000032, A000045, A035513, A097285, A349551, A349552, A349553, A349554.

z1 = 30; zc = 20; zr = 20;
t1 = {1, 2}; Do[t1 = Join[t1, Riffle[Range[n - 1], n], {n}], {n, 3, z1}]; (* A097285 *)
t = Partition[t1, 2];
f[n_] := Fibonacci[n]; r = (1 + Sqrt[5])/2;
s[h_, k_] := Table[h*f[n - 1] + k*f[n], {n, 2, zc}];
w = Table[Join[{h = t[[n]][[1]], k = t[[n]][[2]]}, s[h, k]], {n, 1, zr}]
TableForm[w]  (* A349550 array *)
w1[n_, k_] := w[[n]][[k]];
Table[w1[n - k + 1, k], {n, 13}, {k, n, 1, -1}] // Flatten (* A349550 sequence *)

A097283 Contains exactly once every pair (i,j) satisfying 0 < i < j.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 2, 4, 1, 6, 2, 5, 3, 4, 1, 7, 2, 6, 3, 5, 1, 8, 2, 7, 3, 6, 4, 5, 1, 9, 2, 8, 3, 7, 4, 6, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 1, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 1, 14, 2, 13, 3, 12, 4, 11, 5, 10, 6, 9, 7, 8

Offset: 1

Clark Kimberling, Aug 05 2004

nonn

All pairs (i,j) having i > j occur except those of the form (i,i-1) for i>=3. (Those are included at A097285.)

			Terms are lexically ordered in pairs by sum:
1 2 (sum = 3)
1 3 (sum = 4)
1 4 2 3 (two pairs having sum = 5), etc.

Cf. A097284, A097285.

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

cp's OEIS Frontend

A097286 Rectangular array T by descending antidiagonals: T(n,k) = rank of k-th n in A097285.

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A097287 Rectangular array T by antidiagonals: T(n,k) = rank of n where (n,k) occurs in A097285, for distinct n,k and T(n,n)=0 for all n.

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A349550 Meta-Wythoff array based on A097285: M = (M(n,k)), by downward antidiagonals; every row of M is eventually a row of the Wythoff array, W = A035513, and every row of W is a row of M; see Comments.

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A097283 Contains exactly once every pair (i,j) satisfying 0 < i < j.

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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