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A236829 Number T(n,k) of equivalence classes of ways of placing k 6 X 6 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=6, 0<=k<=floor(n/6)^2, read by rows.

%I A236829 #21 Feb 19 2014 04:36:30
%S A236829 1,1,1,1,1,3,1,3,1,6,1,6,1,10,16,4,1,1,10,51,50,14,1,15,125,293,174,1,
%T A236829 15,239,1065,1234,1,21,423,3075,6124,1,21,672,7371,23259,1,28,1030,
%U A236829 16093,81480,51615,10596,808,31,1
%N A236829 Number T(n,k) of equivalence classes of ways of placing k 6 X 6 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=6, 0<=k<=floor(n/6)^2, read by rows.
%C A236829 The first 13 rows of T(n,k) are:
%C A236829 .\ k  0     1     2     3     4     5     6     7     8     9
%C A236829 n
%C A236829 6     1     1
%C A236829 7     1     1
%C A236829 8     1     3
%C A236829 9     1     3
%C A236829 10    1     6
%C A236829 11    1     6
%C A236829 12    1    10    16     4     1
%C A236829 13    1    10    51    50    14
%C A236829 14    1    15   125   293   174
%C A236829 15    1    15   239  1065  1234
%C A236829 16    1    21   423  3075  6124
%C A236829 17    1    21   672  7371 23259
%C A236829 18    1    28  1030 16093 81480 51615 10596   808    31     1
%H A236829 Christopher Hunt Gribble, <a href="/A236829/a236829.cpp.txt">C++ program</a>
%F A236829 It appears that:
%F A236829 T(n,0) = 1, n>= 6
%F A236829 T(n,1) = (floor((n-6)/2)+1)*(floor((n-6)/2+2))/2, n >= 6
%F A236829 T(c+2*6,2) = A131474(c+1)*(6-1) + A000217(c+1)*floor(6^2/4) + A014409(c+2), 0 <= c < 6, c even
%F A236829 T(c+2*6,2) = A131474(c+1)*(6-1) + A000217(c+1)*floor((6-1)(6-3)/4) + A014409(c+2), 0 <= c < 6, c odd
%F A236829 T(c+2*6,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((6-c-1)/2) + A131941(c+1)*floor((6-c)/2)) + S(c+1,3c+2,3), 0 <= c < 6 where
%F A236829 S(c+1,3c+2,3) =
%F A236829 A054252(2,3),  c = 0
%F A236829 A236679(5,3),  c = 1
%F A236829 A236560(8,3),  c = 2
%F A236829 A236757(11,3), c = 3
%F A236829 A236800(14,3), c = 4
%F A236829 A236829(17,3), c = 5
%e A236829 T(12,3) = 4 because the number of equivalence classes of ways of placing 3 6 X 6 square tiles in a 12 X 12 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
%e A236829 ._________________          _________________
%e A236829 |        |        |        |        |________|
%e A236829 |        |        |        |        |        |
%e A236829 |    .   |    .   |        |    .   |        |
%e A236829 |        |        |        |        |    .   |
%e A236829 |        |        |        |        |        |
%e A236829 |________|________|        |________|        |
%e A236829 |        |        |        |        |________|
%e A236829 |        |        |        |        |        |
%e A236829 |    .   |        |        |    .   |        |
%e A236829 |        |        |        |        |        |
%e A236829 |        |        |        |        |        |
%e A236829 |________|________|        |________|________|
%e A236829 .
%e A236829 ._________________          _________________
%e A236829 |        |        |        |        |        |
%e A236829 |        |________|        |        |        |
%e A236829 |    .   |        |        |    .   |________|
%e A236829 |        |        |        |        |        |
%e A236829 |        |    .   |        |        |        |
%e A236829 |________|        |        |________|    .   |
%e A236829 |        |        |        |        |        |
%e A236829 |        |________|        |        |        |
%e A236829 |    .   |        |        |    .   |________|
%e A236829 |        |        |        |        |        |
%e A236829 |        |        |        |        |        |
%e A236829 |________|________|        |________|________|
%Y A236829 Cf. A054252, A236679, A236560, A236757, A236800, A236865, A236915, A236936, A236939.
%K A236829 tabf,nonn
%O A236829 6,6
%A A236829 _Christopher Hunt Gribble_, Jan 31 2014