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A236757 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4)^2, read by rows.

%I A236757 #25 Feb 18 2014 13:30:37
%S A236757 1,1,1,1,1,3,1,3,1,6,9,3,1,1,6,29,35,14,1,10,75,209,174,1,10,147,765,
%T A236757 1234,1,15,270,2340,7639,6169,1893,242,17,1,1,15,438,5806,34342,79821,
%U A236757 80722,36569,7106,459
%N A236757 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4)^2, read by rows.
%C A236757 The first 10 rows of T(n,k) are:
%C A236757 .\ k  0     1     2     3     4     5     6     7     8     9
%C A236757 n
%C A236757 4     1     1
%C A236757 5     1     1
%C A236757 6     1     3
%C A236757 7     1     3
%C A236757 8     1     6     9     3     1
%C A236757 9     1     6    29    35    14
%C A236757 10    1    10    75   209   174
%C A236757 11    1    10   147   765  1234
%C A236757 12    1    15   270  2340  7639  6169  1893   242    17     1
%C A236757 13    1    15   438  5806 34342 79821 80722 36569  7106   459
%H A236757 Christopher Hunt Gribble, <a href="/A236757/a236757.cpp.txt">C++ program</a>
%F A236757 It appears that:
%F A236757 T(n,0) = 1, n>= 4
%F A236757 T(n,1) = (floor((n-4)/2)+1)*(floor((n-4)/2+2))/2, n >= 4
%F A236757 T(c+2*4,2) = A131474(c+1)*(4-1) + A000217(c+1)*floor(4^2/4) + A014409(c+2), 0 <= c < 4, c even
%F A236757 T(c+2*4,2) = A131474(c+1)*(4-1) + A000217(c+1)*floor((4-1)(4-3)/4) + A014409(c+2), 0 <= c < 4, c odd
%F A236757 T(c+2*4,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((4-c-1)/2) + A131941(c+1)*floor((4-c)/2)) + S(c+1,3c+2,3), 0 <= c < 4 where
%F A236757 S(c+1,3c+2,3) =
%F A236757 A054252(2,3),  c = 0
%F A236757 A236679(5,3),  c = 1
%F A236757 A236560(8,3),  c = 2
%F A236757 A236757(11,3), c = 3
%e A236757 T(8,3) = 3 because the number of equivalence classes of ways of placing 3 4 X 4 square tiles in an 8 X 8 square under all symmetry operations of the square is 3. The portrayal of an example from each equivalence class is:
%e A236757 ._____________          _____________          _____________
%e A236757 |      |      |        |      |______|        |      |      |
%e A236757 |   .  |   .  |        |   .  |      |        |   .  |______|
%e A236757 |      |      |        |      |   .  |        |      |      |
%e A236757 |______|______|        |______|      |        |______|   .  |
%e A236757 |      |      |        |      |______|        |      |      |
%e A236757 |   .  |      |        |   .  |      |        |   .  |______|
%e A236757 |      |      |        |      |      |        |      |      |
%e A236757 |______|______|        |______|______|        |______|______|
%Y A236757 Cf. A054252, A236679, A236560, A236800, A236829, A236865, A236915, A236936, A236939.
%K A236757 nonn,tabf
%O A236757 4,6
%A A236757 _Christopher Hunt Gribble_, Jan 30 2014