A098604 -id:A098604 - cp's OEIS frontend

A103315 Number of minimum dominating sets for the n X n knight graph.

1, 1, 8, 9, 47, 127, 10, 2, 2, 4, 800, 2, 152, 4, 504, 2, 212, 19562

Offset: 1

N. J. A. Sloane, Mar 20 2005, following a suggestion from Lee Morgenstern

In other words, as made explicit in the old name: Sequence A006075 gives minimum number of knights needed to cover an n X n board (i.e., the domination number of the n X n knight graph). This sequence (A103315) gives total number of solutions using A006075(n) knights (compare A006076).

Lee Morgenstern, Knight Domination.
Frank Rubin, Knight coverings for large chessboards, 2000.
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Minimum Dominating Set

Cf. A006075 (domination number of the n X n knight graph).
Cf. A006076 (inequivalent number of minimum dominating sets).
Cf. A098604.

New name from Eric W. Weisstein, Sep 06 2021

A006076 Sequence A006075 gives minimal number of knights needed to cover an n X n board. This sequence gives number of inequivalent solutions using A006075(n) knights.

Original entry on oeis.org

1, 1, 2, 3, 8, 23, 3, 1, 1, 2, 100, 1, 20, 1, 63, 1, 29, 2551

Offset: 1

N. J. A. Sloane

David C. Fisher, On the N X N Knight Cover Problem, Ars Combinatoria 69 (2003), 255-274.
M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 194.
Bernard Lemaire, Knights Covers on N X N Chessboards, J. Recreational Mathematics, Vol. 31-2, 2003, 87-99.
Frank Rubin, Improved knight coverings, Ars Combinatoria 69 (2003), 185-196.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Lee Morgenstern, Knight Domination.
Frank Rubin, Knight coverings for large chessboards, 2000.
Eric Weisstein's World of Mathematics, Knights Problem.

Cf. A006075 (number of solutions), A098604 (rectangular board). A103315 gives the total number of solutions.

a(11) was found in 1973 by Bernard Lemaire. (Philippe Deléham, Jan 06 2004)
a(13)-a(17) from the Morgenstern web site, Nov 08 2004
a(18) from the Morgenstern web site, Mar 20 2005

A110217 Cone C(n,m,k) read by planes and rows, for 1 <= k <= m <= n: minimal number of knights needed to cover a k X m X n board.

Original entry on oeis.org

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

Offset: 1

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jul 17 2005

			Cone starts:
1..2....3......4........5............6.................
...4.8..4.8....4.8......4.8..........4..6
........4.6.6..4.6.6....4.6.6........4..7..6
...............4.6.7.8..4.6.7..8.....4..8..8.12
........................5.6.8.10.13..6..8.10.12.?
.....................................8.11.12..?....

C(n, n, 1) = A006075(n), C(n, k, 1) = A098604(n, k), C(n, n, n) = A110214(n). A110218 gives number of inequivalent ways to cover the board using C(n, m, k) knights, A110219 gives total number.

How many knights with move vector (2, 1, 0) are needed to occupy or attack every field of a k X m X n board? Knights may attack each other.

cp's OEIS Frontend

A103315 Number of minimum dominating sets for the n X n knight graph.

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A006076 Sequence A006075 gives minimal number of knights needed to cover an n X n board. This sequence gives number of inequivalent solutions using A006075(n) knights.

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A110217 Cone C(n,m,k) read by planes and rows, for 1 <= k <= m <= n: minimal number of knights needed to cover a k X m X n board.

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