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A236679 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2)^2, read by rows.

%I A236679 #50 Feb 15 2018 20:36:51
%S A236679 1,1,1,1,1,3,4,2,1,1,3,13,20,14,1,6,37,138,277,273,143,39,7,1,1,6,75,
%T A236679 505,2154,5335,7855,6472,2756,459,1,10,147,1547,10855,50021,153311,
%U A236679 311552,416825,361426,200996,71654,16419,2363,211,11,1,1,10,246,3759,39926,291171
%N A236679 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2)^2, read by rows.
%C A236679 Changing the offset from 2 to 1, this is also the sequence: "Triangle read by rows: T(n,k) is the number of nonequivalent ways to place k non-attacking kings on an n X n board." (For if each king is represented by a 2 X 2 tile with the king in the upper left corner, the kings do not attack each other.) For example, with offset 1, T(4,3) = 20 because there are 20 nonequivalent ways to place 3 kings on a 4 X 4 chessboard so that no king threatens any other. - _Heinrich Ludwig_ and _N. J. A. Sloane_, Dec 21 2016
%C A236679 It appears that rows 2n and 2n-1 both contain n^2 + 1 entries. Rotations and reflections of placements are not counted. If they are to be counted, see A193580. - _Heinrich Ludwig_, Dec 11 2016
%H A236679 Heinrich Ludwig, <a href="/A236679/b236679.txt">Table of n, a(n) for n = 2..107</a>
%H A236679 Christopher Hunt Gribble, <a href="/A236679/a236679.cpp.txt">C++ program</a>
%F A236679 It appears that:
%F A236679 T(n,0) = 1, n>= 2
%F A236679 T(n,1) = (floor((n-2)/2)+1)*(floor((n-2)/2+2))/2, n >= 2
%F A236679 T(c+2*2,2) = A131474(c+1)*(2-1) + A000217(c+1)*floor(2^2/4) + A014409(c+2), 0 <= c < 2, c even
%F A236679 T(c+2*2,2) = A131474(c+1)*(2-1) + A000217(c+1)*floor((2-1)(2-3)/4) + A014409(c+2), 0 <= c < 2, c odd
%F A236679 T(c+2*2,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((2-c-1)/2) + A131941(c+1)*floor((2-c)/2)) + S(c+1,3c+2,3), 0 <= c < 2 where
%F A236679 S(c+1,3c+2,3) =
%F A236679 A054252(2,3),  c = 0
%F A236679 A236679(5,3),  c = 1
%e A236679 T(4,2) = 4 because the number of equivalence classes of ways of placing 2 2 X 2 square tiles in a 4 X 4 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
%e A236679 ._______        _______        _______        _______
%e A236679 | . | . |      | . |___|      | . |   |      |_______|
%e A236679 |___|___|      |___| . |      |___|___|      | . | . |
%e A236679 |       |      |   |___|      |   | . |      |___|___|
%e A236679 |_______|      |_______|      |___|___|      |_______|
%e A236679 The first 6 rows of T(n,k) are:
%e A236679 .\ k  0    1    2    3    4    5    6    7    8    9
%e A236679 n
%e A236679 2     1    1
%e A236679 3     1    1
%e A236679 4     1    3    4    2    1
%e A236679 5     1    3   13   20   14
%e A236679 6     1    6   37  138  277  273  143   39    7    1
%e A236679 7     1    6   75  505 2154 5335 7855 6472 2756  459
%Y A236679 Cf. A054252, A236560, A236757, A236800, A236829, A236865, A236915, A236936, A236939.
%Y A236679 Row sums give A275869.
%Y A236679 Columns 2..7: A279111, A279112, A279113, A279114, A279115, A279116.
%Y A236679 Diagonal T(n,n) is A279117.
%Y A236679 Cf. A193580.
%K A236679 tabf,nonn
%O A236679 2,6
%A A236679 _Christopher Hunt Gribble_, Jan 29 2014
%E A236679 More terms from _Heinrich Ludwig_, Dec 11 2016 (The former entry A279118 from _Heinrich Ludwig_ was merged into this entry by _N. J. A. Sloane_, Dec 21 2016)