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%I #47 Aug 01 2023 15:33:25
%S 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,
%T 81,100,100,100,121,121,121,144,144,144,169,169,169,196,196,196,225,
%U 225,225,256,256,256,289,289,289,324,324,324,361,361,361,400,400
%N Domination number for kings' graph K(n).
%C Also the lower independence number of the n X n knight graph. - _Eric W. Weisstein_, Aug 01 2023
%D John J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004, p. 102.
%H Vincenzo Librandi, <a href="/A075561/b075561.txt">Table of n, a(n) for n = 1..1000</a>
%H Irene Choi, Shreyas Ekanathan, Aidan Gao, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Vaibhav Rastogi, Daniel Sheffield, Michael Yang, Angela Zhao, and Corey Zhao, <a href="https://arxiv.org/abs/2212.01468">The Struggles of Chessland</a>, arXiv:2212.01468 [math.HO], 2022.
%H Matthew D. Kearse and Peter B. Gibbons, <a href="http://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/133chess.pdf">Computational Methods and New Results for Chessboard Problems</a>, Centre for Discrete Mathematics and Theoretical Computer Science, CDMTCS-133, May 2000.
%H Matthew D. Kearse and Peter B. Gibbons, <a href="http://ajc.maths.uq.edu.au/pdf/23/ocr-ajc-v23-p253.pdf">Computational Methods and New Results for Chessboard Problems</a>, Australasian Journal of Combinatorics 23 (2001), 253-284.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingsProblem.html">Kings Problem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LowerIndependenceNumber.html">Lower Independence Number</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F a(n) = floor((n+2)/3)^2. - _Vaclav Kotesovec_, May 13 2012
%F G.f.: -x*(x+1)*(x^2-x+1) / ((x-1)^3*(x^2+x+1)^2). - _Colin Barker_, Oct 06 2014
%F 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
%F Sum_{n>=1} 1/a(n) = Pi^2/2 (A102753). - _Amiram Eldar_, Nov 03 2022
%t Table[Floor[(n + 2)/3]^2, {n, 50}] (* _Vaclav Kotesovec_, May 13 2012 *)
%t LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 1, 1, 4, 4, 4, 9}, 20] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t CoefficientList[Series[(-1 - x^3)/((-1 + x)^3 (1 + x + x^2)^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jun 20 2017 *)
%o (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
%Y Cf. A189889, A075458, A006075, A102753.
%K nonn,easy
%O 1,4
%A _N. J. A. Sloane_, Oct 16 2002
%E More terms added from _Vaclav Kotesovec_, May 13 2012
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