

A279408


Triangle read by rows: T(n,m) (n>=m>=1) = domination number for kings' graph on an n X m toroidal board.


2



1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 4, 4, 2, 2, 2, 4, 4, 4, 3, 3, 3, 5, 5, 6, 7, 3, 3, 3, 6, 6, 6, 8, 8, 3, 3, 3, 6, 6, 6, 9, 9, 9, 4, 4, 4, 7, 7, 8, 10, 11, 12, 14, 4, 4, 4, 8, 8, 8, 11, 11, 12, 15, 15, 4, 4, 4, 8, 8, 8, 12, 12, 12, 16, 16, 16, 5, 5, 5, 9, 9, 10, 13, 14, 15, 18, 19, 20, 22
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OFFSET

1,7


COMMENTS

That is, the minimal number of kings needed to cover an n X m toroidal chessboard so that every square has a king on it, is under attack by a king, or both.
For the usual nontoroidal case, the formula is ceiling(m/3)*ceiling(n/3).


REFERENCES

John J. Watkins, Across the Board: The Mathematics of Chessboard Problem, Princeton University Press, 2004, pages 144149.


LINKS

Indranil Ghosh, Rows 1..100, flattened
Dan Freeman, Chessboard Puzzles Part 4  Other Surfaces and Variations.


FORMULA

T(n,m) = ceiling(max(m*ceiling(n/3), n*ceiling(m/3))/3).


EXAMPLE

T(7,7)=7 can be reached by:
...K...
......K
..K....
.....K.
.K.....
....K..
K......


MATHEMATICA

Flatten[Table[Ceiling[Max[m Ceiling[n/3], n Ceiling[m/3]]/3], {n, 1, 13}, {m, 1, n}]] (* Indranil Ghosh, Mar 09 2017 *)


PROG

(PARI) T(n, m) = ceil(max(m*ceil(n/3), n*ceil(m/3))/3)
for(n=1, 20, for(m=1, n, print1(T(n, m)", "))) \\ Charles R Greathouse IV, Dec 16 2016


CROSSREFS

Cf. A075561, A279402, A279407, A279209.
Sequence in context: A216685 A187184 A301375 * A135592 A022912 A318955
Adjacent sequences: A279405 A279406 A279407 * A279409 A279410 A279411


KEYWORD

nonn,tabl,easy


AUTHOR

Andrey Zabolotskiy, Dec 16 2016


STATUS

approved



