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

1,4

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.

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A189889, A075458, A006075, A102753.

Sequence in context: A200600 A048761 A211547 * A256796 A117405 A013601

Adjacent sequences: A075558 A075559 A075560 * A075562 A075563 A075564

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 16 2002

Extensions

More terms added from Vaclav Kotesovec, May 13 2012

Status

approved