A075561 Domination number for kings' graph K(n).

1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16, 25, 25, 25, 36, 36, 36, 49, 49, 49, 64, 64, 64, 81, 81, 81, 100, 100, 100, 121, 121, 121, 144, 144, 144, 169, 169, 169, 196, 196, 196, 225, 225, 225, 256, 256, 256, 289, 289, 289, 324, 324, 324, 361, 361, 361, 400, 400

Offset: 1

Author

N. J. A. Sloane, Oct 16 2002

Keywords

• nonn
• easy

Comments

Also the lower independence number of the n X n knight graph. - Eric W. Weisstein, Aug 01 2023

References

• John J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004, p. 102.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Irene Choi, Shreyas Ekanathan, Aidan Gao, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Vaibhav Rastogi, Daniel Sheffield, Michael Yang, Angela Zhao, and Corey Zhao, The Struggles of Chessland, arXiv:2212.01468 [math.HO], 2022.
• Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Centre for Discrete Mathematics and Theoretical Computer Science, CDMTCS-133, May 2000.
• Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Australasian Journal of Combinatorics 23 (2001), 253-284.
• Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], 2024. See p. 15.
• Eric Weisstein's World of Mathematics, Domination Number
• Eric Weisstein's World of Mathematics, King Graph
• Eric Weisstein's World of Mathematics, Kings Problem
• Eric Weisstein's World of Mathematics, Lower Independence Number
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A189889, A075458, A006075, A102753.

Programs

Mathematica

Table[Floor[(n + 2)/3]^2, {n, 50}] (* Vaclav Kotesovec, May 13 2012 *)
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 1, 1, 4, 4, 4, 9}, 20] (* Eric W. Weisstein, Jun 20 2017 *)
CoefficientList[Series[(-1 - x^3)/((-1 + x)^3 (1 + x + x^2)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 20 2017 *)

PARI

Vec(-x*(x+1)*(x^2-x+1)/((x-1)^3*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Oct 06 2014

Formula

a(n) = floor((n+2)/3)^2. - Vaclav Kotesovec, May 13 2012

G.f.: -x*(x+1)*(x^2-x+1) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Oct 06 2014

E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 3*x*(3 + x)) + (6*x - 5)*cos(sqrt(3)*x/2) + sqrt(3)*(3 + 2*x)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Oct 17 2022

Sum_{n>=1} 1/a(n) = Pi^2/2 (A102753). - Amiram Eldar, Nov 03 2022

Extensions

More terms added from Vaclav Kotesovec, May 13 2012