A280984 Minimum number of dominoes on an n X n chessboard needed to prevent placement of another domino.

0, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 122, 134, 147, 163, 178, 192, 210, 227, 243, 263, 282, 300, 322, 343, 363

Offset

1,2

Comments

Each domino must cover exactly two adjacent squares of a row or column. Sequence inspired by question for 8 X 8 case in "Minimum Guard Problem" link.

A.k.a. lower matching number of the n X n grid graph. - Eric W. Weisstein, Dec 16 2024

a(n) = 0 for n = 1, a(n) = ceiling(n^2/3) + 1 for n = 19, 22, 23, 25, 26, 28, 29, 31, 32, and a(n) = ceiling(n^2/3) for other n <= 32. - Eric W. Weisstein, Dec 16 2024

Links

Table of n, a(n) for n=1..33.

Andrejs Cibulis and Walter Trump, Domino Exclusion Problem, Baltic J. Modern Computing, Vol. 8 (2020), No. 4, 496-519.

A. Gyárfás, J. Lehel, and Zs. Tuza, Clumsy packing of dominoes, Discrete Mathematics, Volume 71, Issue 1 (1988), 33-46.

Peter Kagey, Minimum number of dominoes on an n X n chessboard to prevent placement of another domino.

Mathematics Stack Exchange user "Manin", Minimum Guard Problem.

Walter Trump, Minimum Domino Packing

Eric Weisstein's World of Mathematics, Grid Graph.

Eric Weisstein's World of Mathematics, Lower Matching Number.

Formula

Proved: a(n) >= A008810(n) for n>1; when n = 0 (mod 3), a(n) = A008810(n). - Andrey Zabolotskiy, Oct 22 2017

a(n) > n^2/3 + n/111 for large n not congruent to 0 (mod 3) [from Gyárfás, Lehel, Tuza]. - Peter Kagey, May 22 2019

Crossrefs

Cf. A008810 (maximum number of L-shaped trominoes with the same orientation in an n X n square, a.k.a. ceil(n^2/3)).

Cf. A378763 (lower matching number of the n X n torus grid graph).

Cf. A379177 (lower matching number of the n X n X n grid graph).

Sequence in context: A213172 A372221 A008810 * A339485 A176893 A144677

Adjacent sequences: A280981 A280982 A280983 * A280985 A280986 A280987

Keyword

nonn,more,changed

Author

Rick L. Shepherd, Jan 11 2017, Aug 06 2017

Extensions

a(10)-a(14) from Lars Blomberg, Aug 08 2017

a(15) from Andrey Zabolotskiy, Oct 20 2017

a(16)-a(17) from Rob Pratt (see the link to Peter Kagey's question) and a(18) added by Andrey Zabolotskiy, Feb 13 2020

a(19)-a(33) from Walter Trump, Jun 14 2020

Status

approved