A269526 -id:A269526 - cp's OEIS frontend

A274528 Square array read by antidiagonals upwards: T(n,k) = A269526(n+1,k+1) - 1, n>=0, k>=0.

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

Offset: 0

Omar E. Pol, Jun 29 2016

This sequence has essentially the same properties as the main sequence A269526, but now involves the nonnegative integers.
This version is important because of the following comment from Allan C. Wechsler, originally contributed to A269526. - N. J. A. Sloane, Jun 30 2016
Sprague-Grundy (Nim) values for a combinatorial game played with two piles of counters. Legal moves consist of removing any positive number of counters from either pile, or removing the same number from both piles, or moving any positive number of counters from the right pile to the left pile. If the Nim-values (as in Sprague-Grundy theory) are written in an array indexed by the number of counters in the two piles, we obtain this array. - Allan C. Wechsler, Jun 29 2016 [corrected by N. J. A. Sloane, Sep 25 2016]
The same sequence arises if we construct a triangle, by reading from left to right in each row, always choosing the smallest nonnegative number which does not produce a duplicate number in any row or diagonal. - N. J. A. Sloane, Jul 02 2016
It appears that the numbers generally appear for the first time in or near the first few rows. - Omar E. Pol, Jul 03 2016

			The corner of the square array begins:
0,  2,  1,  5,  3,  4,  9, 10, 12,  7, 13, 17,
1,  3,  4,  0,  7,  2,  5, 11, 13, 15,  6,
2,  0,  5,  1,  8,  6,  4,  3, 14, 16,
3,  1,  2,  4,  0,  7,  8,  6, 15,
4,  6,  0,  3,  1,  5,  2, 14,
5,  7,  8,  6,  4,  9,  3,
6,  4,  3,  2,  5, 13,
7,  5,  6,  8, 10,
8, 10,  7,  9,
9, 11, 12,
10, 8,
11,

Alois P. Heinz, Antidiagonals 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 illustration of T(n, k) for n = 0..499 and k = 0..499 (where the color is function of T(n, k))
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.

Cf. A269526, A274529, A274534, A274650, A274652.

Columns 1, 2, 3, 4 give A001477, A004443, A274615, A274619.

# From N. J. A. Sloane, Jul 30 2018, based on Heinz's program in A269526
A:= proc(n, k) option remember; local m, s;
         if n=1 and k=1 then 0
       else s:= {seq(A(i, k), i=1..n-1),
                 seq(A(n, j), j=1..k-1),
                 seq(A(n-t, k-t), t=1..min(n, k)-1),
                 seq(A(n+j, k-j), j=1..k-1)};
            for m from 0 while m in s do od; m
         fi
     end:
[seq(seq(A(1+d-k, k), k=1..d), d=1..12)];

A[n_, k_] := A[n, k] = Module[{m, s}, If[n==1 && k==1, 0, s = Join[Table[ A[i, k], {i, 1, n-1}], Table[A[n, j], {j, 1, k-1}], Table[A[n-t, k-t], {t, 1, Min[n, k] - 1}], Table[A[n+j, k-j], {j, 1, k-1}]]; For[m = 0, MemberQ[s, m], m++]; m]];
Table[A[d-k+1, k], {d, 1, 13}, {k, 1, d}] // Flatten (* Jean-François Alcover, May 03 2019, from Maple *)

A274315 First row of infinite Sudoku-type array A269526.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 11, 13, 8, 14, 18, 7, 20, 19, 9, 12, 24, 26, 23, 25, 29, 16, 15, 35, 31, 38, 40, 37, 39, 41, 17, 43, 42, 47, 46, 45, 52, 27, 21, 22, 51, 58, 53, 60, 50, 56, 62, 64, 63, 67, 66, 68, 73, 72, 59, 74, 28, 77, 76, 70, 71, 30, 87, 32, 83, 84, 33, 34, 89, 88, 92, 91, 36, 98, 93, 96

Offset: 1

N. J. A. Sloane, Jun 29 2016

nonn

Conjectured to be a permutation of the natural numbers.
It would be nice to have a formula or recurrence.  Note that the first row of the analogous array corresponding to the Wythoff game, A004482, does have a simple formula.
See A295563 for much more about this sequence. - N. J. A. Sloane, Mar 10 2019

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A004482, A269526, A274316, A274317, A274318, A295563.

A274318 Main diagonal of infinite Sudoku-type array A269526.

Original entry on oeis.org

1, 4, 6, 5, 2, 10, 8, 3, 7, 9, 16, 26, 29, 22, 20, 23, 28, 38, 12, 32, 46, 13, 14, 11, 15, 56, 35, 58, 47, 48, 24, 18, 21, 69, 17, 52, 41, 19, 82, 83, 70, 25, 30, 62, 93, 27, 65, 106, 78, 102, 37, 110, 112, 76, 116, 119, 92, 31, 34, 49, 39, 101, 33, 36, 138

Offset: 1

N. J. A. Sloane, Jun 29 2016

nonn

Conjectured to be a permutation of the natural numbers.

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

Cf. A269526, A274315, A274316, A274317.

A[n_, k_] := A[n, k] = If[n == 1 && k == 1, 1, s = {Table[A[i, k], {i, 1, n - 1}], Table[A[n, j], {j, 1, k - 1}], Table[A[n - t, k - t], {t, 1, Min[n, k] - 1}], Table[A[n + j, k - j], {j, 1, k - 1}]} // Flatten; For[m = 1, True, m++, If[FreeQ[s, m], Return[m]]]];
a[n_] := A[n, n];
Array[a, 65] (* Jean-François Alcover, Jun 10 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jun 29 2016

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

Original entry on oeis.org

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

A274316 Second row of infinite Sudoku-type array A269526.

Original entry on oeis.org

2, 4, 5, 1, 8, 3, 6, 12, 14, 16, 7, 15, 17, 9, 22, 21, 11, 25, 10, 27, 28, 30, 32, 34, 13, 37, 39, 36, 18, 19, 43, 42, 38, 46, 40, 44, 51, 53, 23, 20, 56, 50, 49, 54, 59, 58, 52, 24, 55, 62, 26, 63, 72, 74, 71, 65, 73, 76, 78, 29, 31, 79, 86, 83, 84, 33, 35, 90, 88, 93, 91, 95, 99, 97, 100, 94

Offset: 1

N. J. A. Sloane, Jun 29 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A269526, A274315, A274317, A274318.

A274317 Third row of infinite Sudoku-type array A269526.

Original entry on oeis.org

3, 1, 6, 2, 9, 7, 5, 4, 15, 17, 12, 19, 18, 21, 8, 10, 23, 22, 11, 14, 13, 31, 33, 35, 36, 16, 34, 41, 43, 42, 38, 40, 45, 47, 27, 20, 52, 49, 54, 50, 57, 46, 59, 58, 24, 55, 63, 56, 26, 65, 25, 71, 73, 67, 28, 70, 75, 29, 32, 30, 72, 85, 80, 82, 86, 91, 89, 87, 92, 90, 94, 93, 37, 95, 39, 99

Offset: 1

N. J. A. Sloane, Jun 29 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A269526, A274315, A274316, A274318.

A274530 Sum of the n-th antidiagonal of infinite Sudoku-type array A269526.

Original entry on oeis.org

1, 5, 9, 16, 18, 31, 43, 51, 66, 77, 111, 146, 128, 177, 168, 202, 202, 266, 296, 278, 327, 388, 393, 405, 553, 497, 634, 690, 639, 736, 809, 792, 816, 816, 985, 1006, 1012, 1232, 1245, 1215, 1342, 1435, 1481, 1305, 1629, 1595, 1639, 1693, 1825, 1833, 2061, 2008, 2006, 2407, 2496, 2314, 2525, 2400, 2694, 2723

Offset: 1

Omar E. Pol, Jun 29 2016

nonn

			For n = 6 the sum of all elements of the 6th antidiagonal of the square array A269526 is 6 + 7 + 3 + 2 + 8 + 5 = 31, so a(6) = 31.

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A269526, A274529, A274652.

a(n) = A274652(n-1) + n.

More terms from Alois P. Heinz, Jun 29 2016

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.

A274614 Third column of array in A269526.

Original entry on oeis.org

2, 5, 6, 3, 1, 9, 4, 7, 8, 13, 14, 11, 12, 10, 18, 15, 16, 21, 22, 19, 17, 25, 20, 23, 24, 29, 30, 27, 28, 26, 34, 31, 32, 37, 38, 35, 33, 41, 36, 39, 40, 45, 46, 43, 44, 42, 50, 47, 48, 53, 54, 51, 49, 57, 52, 55, 56, 61, 62, 59, 60, 58, 66, 63, 64, 69, 70

Offset: 1

N. J. A. Sloane, Jun 30 2016

nonn

a(n) = A274615(n-1)+1.

This is a permutation of the natural numbers - see A274615.

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

Cf. A269526, A274615.

More terms from Alois P. Heinz, Jul 01 2016

A275883 Maximal element in n-th antidiagonal of A269526.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 10, 11, 13, 14, 16, 18, 16, 20, 19, 22, 21, 24, 26, 24, 27, 29, 30, 32, 35, 35, 38, 40, 37, 41, 43, 43, 43, 43, 47, 47, 47, 52, 53, 51, 54, 56, 58, 55, 60, 59, 59, 62, 64, 63, 67, 66, 68, 73, 74, 71, 74, 73, 77, 78, 75, 79, 80, 87, 83, 84, 86

Offset: 1

N. J. A. Sloane, Aug 18 2016

nonn

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

Similar to but different from A274529.

Cf. A269526.

cp's OEIS Frontend

A274528 Square array read by antidiagonals upwards: T(n,k) = A269526(n+1,k+1) - 1, n>=0, k>=0.

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A274315 First row of infinite Sudoku-type array A269526.

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A274318 Main diagonal of infinite Sudoku-type array A269526.

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

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A274316 Second row of infinite Sudoku-type array A269526.

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A274317 Third row of infinite Sudoku-type array A269526.

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A274530 Sum of the n-th antidiagonal of infinite Sudoku-type array A269526.

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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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A274614 Third column of array in A269526.

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A275883 Maximal element in n-th antidiagonal of A269526.

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