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A287864 Consider a symmetric pyramid-shaped chessboard with rows of squares of lengths n, n-2, n-4, ..., ending with either 2 or 1 squares; a(n) is the maximal number of mutually non-attacking queens that can be placed on this board.

%I A287864 #48 Mar 29 2024 03:39:20
%S A287864 1,1,1,2,2,3,3,4,4,5,5,6,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,13,
%T A287864 14,14,15,15,16,16,17,17,18,18,19,19,20,20,21,21,21,22,22,23,23
%N A287864 Consider a symmetric pyramid-shaped chessboard with rows of squares of lengths n, n-2, n-4, ..., ending with either 2 or 1 squares; a(n) is the maximal number of mutually non-attacking queens that can be placed on this board.
%C A287864 Since there can be at most one queen per row, for n >= 2, a(n) <= floor(n/2). It would be nice to know how fast this sequence grows. Compare A287867.
%C A287864 If n=2t, the board contains t(t+1) squares; if n=2t+1 it contains (t+1)^2 squares. The number of squares is thus given by the quarter-squares sequence (A002620(n+1)).
%C A287864 The bisection a(2n+1) is A274933. - _Rob Pratt_, Jun 04 2017
%C A287864 For n = 1 to 100, here are the exceptions to the pattern that the values increase by 1 every two steps:
%C A287864 a(1) = a(2) = a(3) = 1
%C A287864 a(12) = a(13) = a(14) = 6
%C A287864 a(27) = a(28) = a(29) = 13
%C A287864 a(44) = a(45) = a(46) = 21
%C A287864 a(59) = a(60) = a(61) = 28
%C A287864 a(74) = a(75) = a(76) = 35
%C A287864 a(89) = a(90) = a(91) = 42. - _Rob Pratt_, Jun 04 2017
%H A287864 Andy Huchala, <a href="/A287864/b287864.txt">Table of n, a(n) for n = 1..212</a> (terms 1..100 from Rob Pratt).
%H A287864 Andy Huchala, <a href="/A287864/a287864.py.txt">Python program</a>.
%e A287864 Q = queen, X = empty square
%e A287864 ---
%e A287864 Q   a(1)=1
%e A287864 ---
%e A287864 QX   a(2)=1
%e A287864 ---
%e A287864 .X.
%e A287864 QXX   a(3)=1
%e A287864 ---
%e A287864 .QX.
%e A287864 XXXQ   a(4)=2
%e A287864 ----
%e A287864 ..X..
%e A287864 .QXX.
%e A287864 XXXQX   a(5)=2
%e A287864 -----
%e A287864 ..QX..
%e A287864 .XXXQ.
%e A287864 XQXXXX   a(6)=3
%e A287864 ------
%e A287864 ...X...
%e A287864 ..QXX..
%e A287864 .XXXQX.
%e A287864 XQXXXXX   a(7)=3
%e A287864 -------
%e A287864 ...QX...
%e A287864 ..XXXQ..
%e A287864 .XQXXXX.
%e A287864 XXXXQXXX   a(8)=4
%e A287864 --------
%e A287864 ....QX....
%e A287864 ...XXXQ...
%e A287864 ..XQXXXX..
%e A287864 .XXXXQXXX.
%e A287864 XXQXXXXXXX   a(10)=5
%e A287864 ----------
%e A287864 .....QX.....
%e A287864 ....XXXQ....
%e A287864 ...XQXXXX...
%e A287864 ..XXXXQXXX..
%e A287864 .XXQXXXXXXX.
%e A287864 XXXXXXXXXQXX   a(12)=6
%e A287864 ------------
%e A287864 ......QX......
%e A287864 .....XXXQ.....
%e A287864 ....XQXXXX....
%e A287864 ...XXXXQXXX...
%e A287864 ..XXQXXXXXXX..
%e A287864 .XXXXXXXXXQXX.
%e A287864 XXXXXXXXXXXXXX   a(14)=6
%e A287864 --------------
%Y A287864 Cf. A002620, A274616, A274933, A287867.
%K A287864 nonn
%O A287864 1,4
%A A287864 _N. J. A. Sloane_, Jun 03 2017
%E A287864 a(15)-a(100) from _Rob Pratt_, Jun 04 2017