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A376732 Triangle read by rows: T(n,k) is the maximum number of squares covered (i.e., attacked) by k independent (i.e., non-attacking) queens on an n X n chessboard.

%I A376732 #30 Nov 07 2024 08:48:30
%S A376732 1,4,0,9,9,0,12,15,16,16,17,23,25,25,25,20,30,35,36,36,36,25,37,45,49,
%T A376732 49,49,49,28,44,55,62,64,64,64,64,33,52,66,76,81,81,81,81,81,36,60,77,
%U A376732 92,100,100,100,100,100,100,41,68,88,104,121,121,121,121,121,121,121
%N A376732 Triangle read by rows: T(n,k) is the maximum number of squares covered (i.e., attacked) by k independent (i.e., non-attacking) queens on an n X n chessboard.
%C A376732 T(2,2) = T(3,3) = 0 indicate that there are no solutions to the n-queens problem when n is 2 or 3.
%H A376732 Mia Muessig, <a href="/A376732/b376732.txt">Table of n, a(n) for n = 1..240</a>
%H A376732 John King, <a href="/A376732/a376732.pdf">Examples for Queens 1,2,3,4,5 up to 11x11</a>.
%H A376732 John King, <a href="/A376732/a376732.jpg">Examples for Queens 6,7,8,9 up to 15x15</a>.
%H A376732 Mia Muessig, <a href="https://gist.github.com/PhoenixSmaug/18deba3e6cf140505b14ec27940038a9">Julia code to compute the sequence</a>
%F A376732 T(n,k) = n^2 for k >= A075324(n), n >= 4.
%e A376732 Triangle begins:
%e A376732   n\k|  1    2    3    4    5    6    7    8    9   10   11   12
%e A376732  ----+-----------------------------------------------------------
%e A376732    1 |  1;
%e A376732    2 |  4,   0;
%e A376732    3 |  9,   9,   0;
%e A376732    4 | 12,  15,  16,  16;
%e A376732    5 | 17,  23,  25,  25,  25;
%e A376732    6 | 20,  30,  35,  36,  36,  36;
%e A376732    7 | 25,  37,  45,  49,  49,  49,  49;
%e A376732    8 | 28,  44,  55,  62,  64,  64,  64,  64;
%e A376732    9 | 33,  52,  66,  76,  81,  81,  81,  81,  81;
%e A376732   10 | 36,  60,  77,  92, 100, 100, 100, 100, 100, 100;
%e A376732   11 | 41,  68,  88, 104, 121, 121, 121, 121, 121, 121, 121;
%e A376732   12 | 44,  76, 101, 120, 134, 142, 144, 144, 144, 144, 144, 144;
%e A376732   13 | 49,  84, 112, 136, 153, 165, 169, 169, 169, 169, 169, ...;
%e A376732   14 | 52,  92, 125, 152, 172, 186, 194, 196, 196, 196, 196, ...;
%e A376732   15 | 57, 100, 136, 168, 193, 209, 221, 224, 225, 225, 225, ...;
%e A376732   16 | 60, 108, 149, 184, 212, 231, 242, 251, 256, 256, 256, ...;
%e A376732   17 | 65, 116, 160, 200, 233, 255, 269, 281, 289, 289, 289, ...;
%e A376732   18 | 68, 124, 173, 216, 252, 277, 294, 310, 322, 324, 324, ...;
%e A376732   ...
%Y A376732 Columns 1..8 are A047461, A374933, A375116, A374934, A374935, A374936, A374937, A374938.
%Y A376732 Cf. A075324, A002567, A075458, A274947.
%K A376732 nonn,tabl
%O A376732 1,2
%A A376732 _John King_, Oct 03 2024
%E A376732 Initial terms by John King and Mia Müßig added by _Mia Muessig_, Oct 05 2024