A274528 -id:A274528 - cp's OEIS frontend

A274529 Total number of distinct numbers in the first n antidiagonals of infinite Sudoku-type array A269526.

1, 3, 4, 6, 6, 8, 10, 11, 13, 14, 16, 18, 18, 20, 20, 22, 22, 24, 26, 26, 27, 29, 30, 32, 35, 35, 38, 40, 40, 41, 43, 43, 43, 43, 47, 47, 47, 52, 53, 53, 54, 56, 58, 58, 60, 60, 60, 62, 64, 64, 67, 67, 68, 73, 74, 74, 74, 74, 77, 78, 78, 79, 80, 87, 87, 87, 87

Offset: 1

Omar E. Pol, Jun 30 2016

nonn

a(n) is also the total number of distinct numbers in the first n antidiagonals of the square array A274528.

			For n = 3, the first three antidiagonals of the square array A269526 are [1], [3, 2], [2, 4, 3]. The first four positive integers are in the list, so a(3) = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A269526, A274528, A274530.
Row lengths of A274534.
A275883 is closely related but a different sequence.

More terms from Alois P. Heinz, Jun 30 2016

A296339 On an infinite 60-degree sector of hexagonal graph paper, fill in cells by antidiagonals so that each contains the least nonnegative integer such that no line of edge-adjacent cells contains a repeated term.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 10 2017

To find the number to enter in a cell (assuming the sector is oriented as in the illustration in the link), look at all the numbers in the cells directly above the cell, in the cells to the "North-West", and in the cells to the "South-West", and take their "mex" (the smallest missing number).

The 0-cells in the array all lie on a perfectly straight line (in contrast to the situation in A274528). Also a(n) = 0 iff n = 2*m*(m+1) for some m.

			The initial rows are as follows (however, this does not show the adjancies between the cells correctly - for that, see the illustration in the link):
   0;
   1,  2;
   2,  0,  1;
   3,  1,  2,  4;
   4,  5,  0,  3,  6;
   5,  3,  4,  6,  7,  8;
   6,  4,  5,  0,  3,  9,  7;
   7,  8,  3,  1,  2,  4,  5,  9;
   8,  6,  7,  2,  0,  1,  9,  4,  3;
   9,  7,  8,  5,  1,  2,  6, 10, 11, 12;
  10, 11,  6,  9,  4,  0, ...
  ...
For example, referring to the illustration in the link and NOT to the triangle here, consider the first 5 in the array. The reason this is 5 is because in the column of cells above that cell we can see 2,0,1, to the NW we see 3, and to the SW we see 4, and the smallest missing number is 5.

Rémy Sigrist, Rows n = 0..200, flattened
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, Colored representation of the first 2^9 rows of the triangle
Rémy Sigrist, PARI program for A296339
N. J. A. Sloane, Illustration of initial rows of the sector.

Two analogs of this for an infinite square chessboard are A269526 (which uses positive numbers) and A274528 (which uses nonnegative numbers).
For the right edge see A296340.
The second column is A004483. - Rémy Sigrist, Dec 11 2017
The third and fourth columns are A004482 and A298801.
See also A274820.

ab = Table[0, {13}];
nw = ab;
A296339 = Reap[For[s = 1, s <= Length[ab], s++, sw = 0; For[c = 1, c <= s, c++, x = BitOr[ab[[c]], BitOr[nw[[s-c+1]], sw]]; v = IntegerExponent[x+1, 2]; Sow[v]; p = 2^v; sw += p; ab[[c]] += p; nw[[s-c+1]] += p]]][[2, 1]] (* Jean-François Alcover, Dec 18 2017, after Rémy Sigrist *) (* I changed the first line, which was ab = Table[0, 13];, to make this compatible with older versions of MMA - N. J. A. Sloane, Feb 03 2018 *)

PARI
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See Links section.
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More terms from Rémy Sigrist, Dec 11 2017

A286297 Irregular triangle read by rows: successive rows have lengths 1,3,5,7,..., and are filled in across rows with the smallest nonnegative number such that there is no repeat in any row, column, or diagonal of slope +-1.

Original entry on oeis.org

0, 1, 2, 3, 2, 0, 4, 1, 5, 3, 1, 5, 6, 0, 4, 2, 4, 5, 0, 2, 1, 7, 3, 6, 8, 5, 3, 1, 4, 6, 8, 9, 2, 7, 0, 10, 6, 4, 2, 0, 3, 9, 5, 10, 11, 1, 12, 7, 13, 7, 8, 9, 5, 4, 6, 10, 3, 2, 12, 0, 11, 14, 15, 1, 8, 6, 10, 3, 7, 2, 11, 4, 9, 13, 1, 5, 15, 12, 16, 14, 17

Offset: 0

N. J. A. Sloane, Jun 01 2017

Conjecture: every column or diagonal of slope +-1 is a permutation of the nonnegative numbers.

			Triangle begins:
..........0,
........1,2,3,
......2,0,4,1,5,
....3,1,5,6,0,4,2,
..4,5,0,2,1,7,3,6,8,
5,3,1,4,6,8,9,2,7,0,10,
...

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

Inspired by A274528, A274641, A274650.

More terms from Alois P. Heinz, Jun 01 2017

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,

A295567 Consider numbers k such that A295563(k) > k (see A295566); sequence lists the values A295563(k).

Original entry on oeis.org

2, 5, 9, 10, 12, 13, 17, 19, 18, 23, 25, 22, 24, 28, 34, 30, 37, 39, 36, 38, 40, 42, 41, 46, 45, 44, 51, 50, 57, 52, 59, 49, 55, 61, 63, 62, 66, 65, 67, 72, 71, 58, 73, 76, 75, 69, 70, 86, 82, 83, 88, 87, 91, 90, 97, 92, 95, 99, 105, 104, 93, 102, 108, 98, 106, 112, 110, 111, 109, 116, 121, 117, 114

Offset: 1

N. J. A. Sloane, Nov 29 2017

nonn

Do the ratios A295567(k)/A295566(k) converge and if so what is the limit?

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

Cf. A269526, A274528, A295563, A295564, A295565, A295566,

A274534 Irregular triangle read by rows: T(n,k) = total number of k's in the first n antidiagonals of infinite Sudoku-type array A269526.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 30 2016

T(n,k) is also the total number of (k-1)'s in the first n antidiagonals of the square array A274528.

			Triangle begins:
1;
1, 1, 1;
1, 2, 2, 1;
2, 2, 2, 2, 1, 1;
3, 3, 2, 3, 2, 2;
3, 4, 3, 3, 3, 3, 1, 1;
4, 4, 4, 3, 4, 3, 2, 2, 1, 1;
5, 4, 4, 4, 5, 4, 3, 3, 2, 1, 1;
5, 5, 4, 5, 6, 5, 4, 4, 3, 1, 1, 1, 1;
5, 5, 5, 6, 7, 6, 5, 5, 4, 2, 2, 1, 1, 1;
5, 5, 6, 6, 7, 7, 6, 6, 5, 3, 3, 2, 1, 2, 1, 1;
5, 5, 6, 7, 7, 7, 7, 6, 6, 4, 4, 3, 2, 3, 2, 2, 1, 1;
5, 5, 7, 8, 7, 8, 8, 7, 7, 5, 5, 4, 3, 4, 3, 3, 1, 1;
...
For n = 3, the first three antidiagonals of the square array A269526 are [1], [3, 2], [2, 4, 3]. There are only one 1, two 2's, two 3's and only one 4, so the third row of the triangle is [1, 2, 2, 1].

Alois P. Heinz, Rows n = 1..175, flattened

Cf. A269526, A274528.
Row sums give A000217, n >= 1.
Row lengths give A274529.

cp's OEIS Frontend

A274529 Total number of distinct numbers in the first n antidiagonals of infinite Sudoku-type array A269526.

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A296339 On an infinite 60-degree sector of hexagonal graph paper, fill in cells by antidiagonals so that each contains the least nonnegative integer such that no line of edge-adjacent cells contains a repeated term.

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A286297 Irregular triangle read by rows: successive rows have lengths 1,3,5,7,..., and are filled in across rows with the smallest nonnegative number such that there is no repeat in any row, column, or diagonal of slope +-1.

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

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A295567 Consider numbers k such that A295563(k) > k (see A295566); sequence lists the values A295563(k).

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A274534 Irregular triangle read by rows: T(n,k) = total number of k's in the first n antidiagonals of infinite Sudoku-type array A269526.

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