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A379726 Minimum number of kings that must be placed on an n X n chessboard such that each square is attacked or occupied by at least two kings.

%I A379726 #45 Mar 31 2025 07:00:31
%S A379726 2,3,8,8,10,18,18,21,32,32,36,50,50,55,72,72,78,98,98,105,128,128,136,
%T A379726 162,162,171,200,200,210,242,242,253,288,288,300,338,338,351,392,392,
%U A379726 406,450,450,465,512,512,528,578,578,595,648,648,666,722,722,741,800,800,820,882,882,903,968,968,990,1058,1058,1081,1152,1152,1176,1250,1250,1275,1352,1352,1378,1458,1458,1485,1568,1568,1596,1682,1682,1711,1800,1800,1830,1922,1922,1953,2048,2048,2080,2178,2178,2211,2312
%N A379726 Minimum number of kings that must be placed on an n X n chessboard such that each square is attacked or occupied by at least two kings.
%C A379726 At most one king can be placed on each square.
%C A379726 Every third term is conjectured to be A014105. Other terms are A001105. A093353 is conjectured to be this sequence with repeated terms removed.
%C A379726 The above conjectures are true (see Beveridge link). - _Colin Beveridge_, Jan 13 2025
%H A379726 Rob Pratt, <a href="/A379726/b379726.txt">Table of n, a(n) for n = 2..100</a>
%H A379726 Colin Beveridge, <a href="/A379726/a379726.pdf">Proof of the case where n is a multiple of 3</a>
%H A379726 Matthew Scroggs, <a href="https://www.mscroggs.co.uk/puzzles/advent2024/23">December 23</a>
%H A379726 Matthew Scroggs, <a href="https://www.mscroggs.co.uk/blog/114">Friendly squares</a>
%H A379726 Matthew Scroggs, <a href="https://github.com/mscroggs/oeis/blob/main/a379726.py">Python code to compute A379726</a>
%H A379726 Puzzling StackExchange, <a href="https://puzzling.stackexchange.com/q/129769/65277">Minimum Number of Squares to Color</a>
%H A379726 Dominic McCarty, <a href="/A379726/a379726.txt">Illustration of a(n) for n = 2..100</a>
%F A379726 If n is not a multiple of 3, a(n) = 2*floor((n+2)/3)^2.
%F A379726 If n is a multiple of 3, it is conjectured that a(n)=2*(n/3)^2+n/3.
%F A379726 The above conjectures are true (see Beveridge link). - _Colin Beveridge_, Jan 13 2025
%e A379726 For a 3 by 3 chessboard, the three kings could be placed like this (where o is an empty square and k is a king):
%e A379726    ooo
%e A379726    kkk
%e A379726    ooo
%e A379726 For a 4 by 4 chessboard, the kings could be placed like this:
%e A379726    oooo
%e A379726    kkkk
%e A379726    okko
%e A379726    okko
%Y A379726 Cf. A001105, A014105, A075561, A093353, A379759, A379766.
%K A379726 nonn
%O A379726 2,1
%A A379726 _Matthew Scroggs_, Dec 31 2024
%E A379726 a(15)-a(100) via integer linear programming by _Rob Pratt_, Jan 02 2025