A269526 -id:A269526 - cp's OEIS frontend

A274080 Table read by rows: row n gives all numbers less than n in the same row, column, or diagonal as n in the natural numbers read by antidiagonals.

1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 3, 4, 5, 1, 2, 4, 2, 3, 4, 5, 7, 2, 3, 5, 6, 7, 8, 1, 3, 6, 7, 8, 9, 1, 2, 4, 7, 3, 4, 5, 7, 8, 11, 1, 4, 5, 6, 8, 9, 11, 12, 2, 5, 6, 9, 10, 11, 12, 13, 1, 3, 6, 10, 11, 12, 13, 14, 1, 2, 4, 7, 11, 3, 5, 7, 8, 11, 12, 16, 2, 6, 7

Offset: 1

Peter Kagey, Jun 09 2016

			A000027 read by antidiagonals is:
1 2 4 7
3 5 8
6 9
...
Thus:
Row 1: []
Row 2: [1]
Row 3: [1, 2]
Row 4: [1, 2]
Row 5: [1, 2, 3, 4]
Row 6: [1, 3, 4, 5]
Row 7: [1, 2, 4]
Row 8: [2, 3, 4, 5, 7]
Row 9: [2, 3, 5, 6, 7, 8]

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027, A269526, A273823, A273824, A273825, A274079.

import Data.List (sort, nub)
a274080 n = a274080_list !! (n - 1)
a274080_list = concatMap a274080_row [1..]
a274080_tabf = map a274080_row [1..]
a274080_row n = nub $ sort $ concatMap (\f -> f n) [a274079_row, a273825_row, a273824_row, a273823_row]

nn = 18; t = Table[(n^2 - n)/2 + Accumulate@ Range[n - 1, Ceiling[(Sqrt[9 + 8 nn] - 3)/2]] + 1, {n, Ceiling[(Sqrt[9 + 8 nn] - 3)/2] + 1}]; Table[Function[a, Function[p, Most@ Union@ Flatten@ {Map[a[[#1, #2]] & @@ # &, Most@ NestWhileList[# - 1 &, First@ p, ! MemberQ[#, 0] &]], Range[SelectFirst[Reverse@ Join[{0}, First@ t], n >= # &], n - 1], Transpose[a][[ p[[1, 2]] ]], a[[ p[[1, 1]] ]]}]@ Position[a, n]]@ Array[t[[#1, #2]] &, First@ Position[t, n]], {n, nn}] // Flatten (* Michael De Vlieger, Jun 29 2016, Version 10 *)

A273823 Table read by rows: the n-th row is the list of numbers to the left of n in the natural numbers read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 5, 3, 6, 7, 4, 2, 1, 8, 5, 3, 9, 6, 10, 11, 7, 4, 2, 1, 12, 8, 5, 3, 13, 9, 6, 14, 10, 15, 16, 11, 7, 4, 2, 1, 17, 12, 8, 5, 3, 18, 13, 9, 6, 19, 14, 10, 20, 15, 21, 22, 16, 11, 7, 4, 2, 1, 23, 17, 12, 8, 5, 3, 24, 18, 13, 9, 6, 25, 19, 14

Offset: 1

Peter Kagey, Jun 08 2016

			A000027 read by antidiagonals is:
1 2 4 7
3 5 8
6 9
...
Thus:
Row 1: []
Row 2: [1]
Row 3: []
Row 4: [2, 1]
Row 5: [3]
Row 6: []
Row 7: [4, 2, 1]
Row 8: [5, 3]
Row 9: [6]

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027, A025581 (row length), A269526, A271439, A273824, A273825, A274079, A274080.

a273823 n = genericIndex a273823_list (n - 1)
a273823_list = concatMap a273823_row [1..]
a273823_tabf = map a273823_row [1..]
a273823_row n
  | a_i == 0  = []
  | otherwise = a_i : a273823_row a_i where
    a_i = a271439 n

nn = 32; t = Table[(n^2 - n)/2 + Accumulate@ Range[n - 1, Ceiling[(Sqrt[9 + 8 nn] - 3)/2]] + 1, {n, Ceiling[(Sqrt[9 + 8 nn] - 3)/2] + 1}]; Table[Reverse@ Take[t[[#1]], #2 - 1] & @@ Flatten@ Position[t, n], {n, nn}] // Flatten (* Michael De Vlieger, Jun 10 2016 *)

A273824 Table read by rows: the n-th row is the list of numbers above n in the table of natural numbers read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 6, 3, 1, 7, 8, 4, 9, 5, 2, 10, 6, 3, 1, 11, 12, 7, 13, 8, 4, 14, 9, 5, 2, 15, 10, 6, 3, 1, 16, 17, 11, 18, 12, 7, 19, 13, 8, 4, 20, 14, 9, 5, 2, 21, 15, 10, 6, 3, 1, 22, 23, 16, 24, 17, 11, 25, 18, 12, 7, 26, 19, 13, 8, 4, 27, 20, 14, 9, 5

Offset: 1

Peter Kagey, Jun 08 2016

			A000027 read by antidiagonals is:
1 2 4 7
3 5 8
6 9
...
Thus:
Row 1: []
Row 2: []
Row 3: [1]
Row 4: []
Row 5: [2]
Row 6: [3, 1]
Row 7: []
Row 8: [4]
Row 9: [5, 2]

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027, A269526, A271439, A273823, A273825, A274079, A274080.

a273824 n = genericIndex a273824_list (n - 1)
a273824_list = concatMap a273824_row [1..]
a273824_tabf = map a273824_row [1..]
a273824_row n
  | a_i == 0  = []
  | otherwise = a_i : a273824_row a_i where
    a_i = a271439 (n - 1)

nn = 35; t = Transpose@ Table[(n^2 - n)/2 + Accumulate@ Range[n - 1, n + Ceiling[(Sqrt[9 + 8 nn] - 3)/2]] + 1, {n, Ceiling[(Sqrt[9 + 8 nn] - 3)/2] + 1}]; Table[Reverse@ Take[t[[#1]], #2 - 1] & @@ Flatten@ Position[t, n], {n, nn}] // Flatten (* Michael De Vlieger, Jun 10 2016 *)

A273825 Table read by rows: the n-th row is the list of numbers diagonally up and to the left of n in the natural numbers read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 6, 7, 8, 2, 9, 3, 10, 11, 12, 4, 13, 5, 1, 14, 6, 15, 16, 17, 7, 18, 8, 2, 19, 9, 3, 20, 10, 21, 22, 23, 11, 24, 12, 4, 25, 13, 5, 1, 26, 14, 6, 27, 15, 28, 29, 30, 16, 31, 17, 7, 32, 18, 8, 2, 33, 19, 9, 3, 34, 20, 10, 35, 21, 36, 37, 38, 22

Offset: 1

Peter Kagey, Jun 08 2016

			A000027 read by antidiagonals is:
1 2 4 7
3 5 8
6 9
...
Thus:
Row 1: []
Row 2: []
Row 3: []
Row 4: []
Row 5: [1]
Row 6: []
Row 7: []
Row 8: [2]
Row 9: [3]

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027, A269526, A271439, A273823, A273824, A274079, A274080.

a273825 n = genericIndex a273825_list (n - 1)
a273825_list = concatMap a273825_row [1..]
a273825_tabf = map a273825_row [1..]
a273825_row n
  | a_i == 0  = []
  | otherwise = a_i : a273825_row a_i where
    a_i = a271439 $ a271439 (n - 1)

nn = 58; t = Table[(n^2 - n)/2 + Accumulate@ Range[n - 1, Ceiling[(Sqrt[9 + 8 nn] - 3)/2]] + 1, {n, Ceiling[(Sqrt[9 + 8 nn] - 3)/2] + 1}]; Table[Rest@ Map[t[[#1, #2]] & @@ # &, Most@ NestWhileList[# - 1 &, #, ! MemberQ[#, 0] &]] &@ First@ Position[t, n], {n, nn}] // Flatten (* Michael De Vlieger, Jun 29 2016 *)

A274079 Table read by rows: the n-th row is the list of numbers diagonally up and to the right of n in the natural numbers read by antidiagonals.

Original entry on oeis.org

2, 4, 5, 4, 7, 8, 7, 9, 8, 7, 11, 12, 11, 13, 12, 11, 14, 13, 12, 11, 16, 17, 16, 18, 17, 16, 19, 18, 17, 16, 20, 19, 18, 17, 16, 22, 23, 22, 24, 23, 22, 25, 24, 23, 22, 26, 25, 24, 23, 22, 27, 26, 25, 24, 23, 22, 29, 30, 29, 31, 30, 29, 32, 31, 30, 29, 33, 32

Offset: 1

Peter Kagey, Jun 09 2016

			A000027 read by antidiagonals is:
1 2 4 7
3 5 8
6 9
...
Thus:
Row 1: []
Row 2: []
Row 3: [2]
Row 4: []
Row 5: [4]
Row 6: [5, 4]
Row 7: []
Row 8: [7]
Row 9: [8, 7]

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027, A269526, A273823, A273824, A273825, A274080.

a274079 n = a274079_list !! (n - 1)
a274079_list = concatMap a274079_row [1..]
a274079_tabf = map a274079_row [1..]
a274079_row n = [n-1, n-2..n - a002262 (n - 1)]

Table[Reverse@ Range[SelectFirst[Reverse@ #, # < n &] + 1, n - 1] - 1, {n, 2, 35}] &[Accumulate@ Range[0, 15] + 1] // Flatten (* Michael De Vlieger, Jun 10 2016, Version 10 *)

A288530 Triangle read by rows in reverse order: T(n,k), (0 <= k <= n), in which each term is the least nonnegative integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

Original entry on oeis.org

0, 1, 2, 2, 0, 3, 3, 1, 4, 5, 4, 5, 0, 2, 1, 5, 3, 1, 4, 6, 7, 6, 4, 2, 0, 3, 8, 9, 7, 8, 9, 1, 4, 5, 10, 6, 8, 6, 5, 3, 0, 2, 7, 9, 11, 9, 7, 10, 11, 2, 6, 8, 12, 3, 4, 10, 11, 6, 8, 7, 0, 12, 13, 14, 5, 15, 11, 9, 7, 10, 5, 1, 6, 8, 15, 16, 12, 13, 12, 10, 8, 6, 9, 3, 0, 11, 5, 7, 13, 14, 16

Offset: 0

Omar E. Pol, Jun 10 2017

Note that the n-th row of this triangle is constructed from right to left, starting at the column n and ending at the column 0.
Theorem 1: the middle diagonal gives A000004, the all-zeros sequence.
Theorem 2: all zeros are in the middle diagonal.
For the proofs of the theorems 1 and 2 see the proofs of the theorems 1 and 2 of A274650, because this is essentially the same problem.
Conjecture 3: every column is a permutation of the nonnegative integers.
Conjecture 4: every diagonal is a permutation of the right border which gives the nonnegative integers.

			Note that every row of the triangle is constructed from right to left, so the sequence is 0, 1, 2, 2, 0, 3, ... (see below):
0,
2,   1,
3,   0,  2,
5,   4,  1,  3,
1,   2,  0,  5,  4,                      Every row is constructed
7,   6,  4,  1,  3,  5,              <---   from right to left.
9,   8,  3,  0,  2,  4,  6,
6,  10,  5,  4,  1,  9,  8,  7,
11,  9,  7,  2,  0,  3,  5,  6,  8,
4,   3, 12,  8,  6,  2, 11, 10,  7,  9,
15,  5, 14, 13, 12,  0,  7,  8,  6, 11, 10,
13, 12, 16, 15,  8,  6,  1,  5, 10,  7,  9, 11,
16, 14, 13,  7,  5, 11,  0,  3,  9,  6,  8, 10, 12,
...
The triangle may be reformatted as an isosceles triangle so that the all-zeros sequence (A000004) appears in the central column (but note that this is NOT the way the triangle is constructed!):
.
.              0,
.            2,  1,
,          3,  0,  2,
.        5,  4,  1,  3,
.      1,  2,  0,  5,  4,
.    7,  6,  4,  1,  3,  5,
.  9,  8,  3,  0,  2,  4,  6,
...
Also the triangle may be reformatted for reading from left to right:
.
.                           0;
.                       1,  2;
.                   2,  0,  3;
.               3,  1,  4,  5;
.           4,  5,  0 , 2,  1;
.       5,  3,  1,  4,  6,  7;
.   6,  4,  2,  0,  3,  8,  9;
...

Alois P. Heinz, Rows n = 0..200, flattened

Middle diagonal gives A000004.
Right border gives A001477.
Indices of the zeros are in A046092.
Cf. A288531 is the same triangle but with 1 added to every entry.
Other sequences of the same family are A269526, A274528, A274650, A274651, A274820, A274821, A286297.

T(n,k) = A288531(n+1, k+1) - 1.

T(n,n) = n.

A288531 Triangle read by rows in reverse order: T(n,k), (1<=k<=n), in which each term is the least positive integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

Original entry on oeis.org

1, 2, 3, 3, 1, 4, 4, 2, 5, 6, 5, 6, 1, 3, 2, 6, 4, 2, 5, 7, 8, 7, 5, 3, 1, 4, 9, 10, 8, 9, 10, 2, 5, 6, 11, 7, 9, 7, 6, 4, 1, 3, 8, 10, 12, 10, 8, 11, 12, 3, 7, 9, 13, 4, 5, 11, 12, 7, 9, 8, 1, 13, 14, 15, 6, 16, 12, 10, 8, 11, 6, 2, 7, 9, 16, 17, 13, 14, 13, 11, 9, 7, 10, 4, 1, 12, 6, 8, 14, 15, 17

Offset: 1

Omar E. Pol, Jun 10 2017

Note that the n-th row of this triangle is constructed from right to left, starting at the column n and ending at the column 1.
Theorem 1: the middle diagonal gives A000012, the all 1's sequence.
Theorem 2: all 1's are in the middle diagonal.
For the proofs of the theorems 1 and 2 see the proofs of the theorems 1 and 2 of A274650, because this is essentially the same problem.
Conjecture 3: every column is a permutation of the positive integers.
Conjecture 4: every diagonal is a permutation of the right border which gives the positive integers.

			Note that every row of the triangle is constructed from right to left, so the sequence is 1, 2, 3, 3, 1, 4,... (see below):
1,
3,   2,
4,   1,  3,
6,   5,  2,  4,
2,   3,  1,  6,  5,                      Every row is constructed
8,   7,  5,  2,  4,  6,              <---   from right to left.
10,  9,  4,  1,  3,  5,  7,
7,  11,  6,  5,  2, 10,  9,  8,
12, 10,  8,  3,  1,  4,  6,  7,  9,
5,   4, 13,  9,  7,  3, 12, 11,  8, 10,
16,  6, 15, 14, 13,  1,  8,  9,  7, 12, 11,
14, 13, 17, 16,  9,  7,  2,  6, 11,  8, 10, 12,
17, 15, 14,  8,  6, 12,  1,  4, 10,  7,  9, 11, 13,
...
The triangle may be reformatted as an isosceles triangle so that the all 1's sequence (A000012) appears in the central column (but note that this is NOT the way the triangle is constructed!):
.
.                1,
.              3,  2,
.            4,  1,  3,
.          6,  5,  2,  4,
.        2,  3,  1,  6,  5,
.      8,  7,  5,  2,  4,  6,
.   10,  9,  4,  1,  3,  5,  7,
...
Also the triangle may be reformatted for reading from left to right:
.
.                           1;
.                       2,  3;
.                   3,  1,  4;
.               4,  2,  5,  6;
.           5,  6,  1 , 3,  2;
.       6,  4,  2,  5,  7,  8;
.   7,  5,  3,  1,  4,  9, 10;
...

Alois P. Heinz, Rows n = 1..201, flattened

Middle diagonal gives A000012.
Right border gives A000027.
Indices of the 1's are in A001844.
Cf. A288530 is the same triangle but with every entry minus 1.
Other sequences of the same family are A269526, A274528, A274650, A274651, A274820, A274821, A286297.

T(n,k) = A288530(n-1,k-1) + 1.

T(n,n) = n.

A295564 Numbers k such that A295563(k) <= k.

Original entry on oeis.org

0, 2, 4, 5, 9, 12, 15, 16, 22, 23, 31, 38, 39, 40, 57, 62, 64, 67, 68, 73, 90, 99, 101, 107, 110, 117, 126, 133, 143, 155, 160, 162, 165, 166, 171, 175, 177, 182, 194, 198, 207, 208, 213, 224, 236, 241, 245, 246, 248, 260, 261, 265, 266, 285, 291, 293, 297, 298, 304, 311, 328, 329, 332, 337, 338, 341

Offset: 1

N. J. A. Sloane, Nov 29 2017

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..15000

Cf. A269526, A274528, A295563, A295565, A295566, A295567,

A295565 Consider numbers k such that A295563(k) <= k (see A295564); sequence lists the values A295563(k).

Original entry on oeis.org

0, 1, 3, 4, 7, 6, 8, 11, 15, 14, 16, 26, 20, 21, 27, 29, 31, 32, 33, 35, 43, 47, 48, 54, 53, 56, 60, 64, 68, 74, 77, 78, 80, 79, 81, 84, 85, 89, 94, 96, 101, 100, 103, 107, 113, 115, 118, 119, 120, 126, 127, 128, 129, 139, 141, 142, 143, 144, 147, 151, 158, 160, 159, 164, 163, 167, 165, 175, 177, 180

Offset: 1

N. J. A. Sloane, Nov 29 2017

nonn

Do the ratios A295565(k)/A295564(k) converge and if so what is the limit?

Rémy Sigrist, Table of n, a(n) for n = 1..15000

Cf. A269526, A274528, A295563, A295564, A295566, A295567,

A295566 Numbers k such that A295563(k) > k.

Original entry on oeis.org

1, 3, 6, 7, 8, 10, 11, 13, 14, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 63, 65, 66, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98

Offset: 1

N. J. A. Sloane, Nov 29 2017

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..50000

Cf. A269526, A274528, A295563, A295564, A295565, A295567,

cp's OEIS Frontend

A274080 Table read by rows: row n gives all numbers less than n in the same row, column, or diagonal as n in the natural numbers read by antidiagonals.

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A273823 Table read by rows: the n-th row is the list of numbers to the left of n in the natural numbers read by antidiagonals.

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A273824 Table read by rows: the n-th row is the list of numbers above n in the table of natural numbers read by antidiagonals.

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A273825 Table read by rows: the n-th row is the list of numbers diagonally up and to the left of n in the natural numbers read by antidiagonals.

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A274079 Table read by rows: the n-th row is the list of numbers diagonally up and to the right of n in the natural numbers read by antidiagonals.

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A288530 Triangle read by rows in reverse order: T(n,k), (0 <= k <= n), in which each term is the least nonnegative integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

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A288531 Triangle read by rows in reverse order: T(n,k), (1<=k<=n), in which each term is the least positive integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

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A295564 Numbers k such that A295563(k) <= k.

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A295565 Consider numbers k such that A295563(k) <= k (see A295564); sequence lists the values A295563(k).

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