cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A279402 Domination number for queen graph on an n X n toroidal board.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 7, 7, 5, 8, 9, 8, 10, 10, 7, 11
Offset: 1

Views

Author

Andrey Zabolotskiy, Dec 11 2016

Keywords

Comments

That is, the minimal number of queens needed to cover an n X n toroidal chessboard so that every square either has a queen on it, or is under attack by a queen, or both.
Row lengths of the triangle A279403.
All dominating sets are translation-invariant on the torus.
a(4*n) <= 2*n.
a(n) <= A075458(n).

Examples

			The minimal dominating set for the queens' graph on a 15 X 15 toroidal board is:
...............
..........Q....
...............
...............
.Q.............
...............
...............
.......Q.......
...............
...............
.............Q.
...............
...............
....Q..........
...............
Hence a(15) = 5.

References

  • John J. Watkins, Across the Board: The Mathematics of Chessboard Problem, Princeton University Press, 2004, pp. 139-140.

Links

Crossrefs

Cf. A075458, A274138, A279403, A279404, A279405, A279406, A279407, A279408, A279409.

Formula

a(3*n) = n if n == 1, 5, 7, 11 (mod 12);
a(3*n) = n+1 if n == 2, 10 (mod 12);
a(3*n) = n+2 otherwise.
I.e., a(3*n) = 2*n - A085801(n).

Extensions

a(16)-a(22) from Andy Huchala, Mar 04 2024

A279405 Peaceable coexisting armies of queens on a torus: the maximum number m such that m white queens and m black queens can coexist on an n X n toroidal chessboard without attacking each other.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 8, 7, 12, 10, 18
Offset: 1

Views

Author

Andrey Zabolotskiy, Dec 11 2016

Keywords

Comments

a(n) <= A250000(n).
a(n) is maximal m such that A279406(n,m) >= m.

Examples

			A solution for n=6:
......
.W...W
...B..
..B.B.
...B..
.W...W

Links

Crossrefs

Cf. A250000, A279406.

Extensions

a(10)-a(12) from Andy Huchala, Mar 10 2024

A279403 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= n^2) = minimal number of squares not attacked by k queens on an n X n toroidal board, with trailing zeros truncated.

Original entry on oeis.org

1, 4, 9, 16, 4, 25, 8, 2, 36, 16, 4, 49, 24, 11, 4, 64, 36, 16, 6, 81, 48, 27, 12, 3, 100, 64, 36, 19, 4, 121, 80, 51, 29, 13, 144, 100, 64, 39, 16, 6, 169, 120, 83, 53, 29, 8, 2, 196, 144, 100, 67, 36, 18, 8, 225, 168, 223, 82, 41, 256, 196, 144, 103, 64, 40
Offset: 1

Views

Author

Andrey Zabolotskiy, Dec 11 2016

Keywords

Comments

Row lengths are A279402.

Examples

			The triangle begins:
1 (0)
4 (0, 0, 0, 0)
9 (0, 0, ...)
16 4 (0, 0, ...)
25 8 2
36 16 4
49 24 11 4
64 36 16 6
81 48 27 12 3
100 64 36 19 4
121 80 51 29 13
144 100 64 39 16 6
169 120 83 53 29 8 2
196 144 100 67 36 18 8
225 168 123 82 41
256 196 144 103 64 40 ...

Crossrefs

Cf. A279402, A279406.

Formula

T(n,0) = A000290(n).
T(n,1) = A000290(n)-A047461(n) = A137932(n-1).
T(n,2) = A248825(n-4) for n >= 6.
Showing 1-3 of 3 results.

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