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Triangle T(n,k) read by rows, for 1 <= k <= n: minimal number of knights needed to cover a k X n board.
3

%I #24 Jun 25 2019 01:35:46

%S 1,2,4,3,4,4,4,4,4,4,5,4,4,4,5,6,4,4,4,6,8,7,6,6,6,7,8,10,8,8,8,8,8,8,

%T 11,12,9,8,8,8,8,10,12,13,14,10,8,8,8,9,12,14,14,15,16,11,8,8,8,10,12,

%U 15,16,17,19,21,12,8,8,8,10,12,16,16,18,20,22,24,13,10,10,10,12,14

%N Triangle T(n,k) read by rows, for 1 <= k <= n: minimal number of knights needed to cover a k X n board.

%C How many knights are needed to occupy or attack every square of a k X n board?

%C I do not know how many of these numbers have been proved to be optimal. - _N. J. A. Sloane_, Nov 08 2004

%H Lee Morgenstern, <a href="https://web.archive.org/web/20070102070601/http://home.earthlink.net/~morgenstern/">Knight Domination</a>.

%H Frank Rubin, <a href="http://www.contestcen.com/knight.htm">Knight coverings for large chessboards</a>, 2000.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnightsProblem.html">Knights Problem</a>.

%e Triangle (with rows n >= 1 and columns k >= 1) begins as follows:

%e 1

%e 2 4

%e 3 4 4

%e 4 4 4 4

%e 5 4 4 4 5

%e 6 4 4 4 6 8

%e 7 6 6 6 7 8 10

%e ...

%Y See A006075 for the n X n case (the main diagonal). A006076 gives number of ways to cover an n X n board using the minimal number of knights.

%K nonn,tabl,nice

%O 1,2

%A _N. J. A. Sloane_

%E Morgenstern's table extends a long way beyond what is shown here.