cp's OEIS Frontend

A236560 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3)^2, read by rows.

%I A236560 #33 Feb 18 2014 13:30:31
%S A236560 1,1,1,1,1,3,1,3,6,2,1,1,6,21,29,14,1,6,53,161,174,1,10,111,665,1713,
%T A236560 1549,608,107,11,1,1,10,201,1961,9973,24267,29437,17438,4756,459
%N A236560 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3)^2, read by rows.
%C A236560 The first 8 rows of T(n,k) are:
%C A236560 .\ k  0     1     2     3     4     5     6     7     8     9
%C A236560 n
%C A236560 3     1     1
%C A236560 4     1     1
%C A236560 5     1     3
%C A236560 6     1     3     6     2     1
%C A236560 7     1     6    21    29    14
%C A236560 8     1     6    53   161   174
%C A236560 9     1    10   111   665  1713  1549   608   107    11     1
%C A236560 10    1    10   201  1961  9973 24267 29437 17438  4756   459
%H A236560 Christopher Hunt Gribble, <a href="/A236560/a236560.cpp.txt">C++ program</a>
%F A236560 It appears that:
%F A236560 T(n,0) = 1, n>= 3
%F A236560 T(n,1) = (floor((n-3)/2)+1)*(floor((n-3)/2+2))/2, n >= 3
%F A236560 T(c+2*3,2) = A131474(c+1)*(3-1) + A000217(c+1)*floor(3^2/4) + A014409(c+2), 0 <= c < 3, c even
%F A236560 T(c+2*3,2) = A131474(c+1)*(3-1) + A000217(c+1)*floor((3-1)(3-3)/4) + A014409(c+2), 0 <= c < 3, c odd
%F A236560 T(c+2*3,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((3-c-1)/2) + A131941(c+1)*floor((3-c)/2)) + S(c+1,3c+2,3), 0 <= c < 3 where
%F A236560 S(c+1,3c+2,3) =
%F A236560 A054252(2,3),  c = 0
%F A236560 A236679(5,3),  c = 1
%F A236560 A236560(8,3),  c = 2
%e A236560 T(6,2) = 6 because the number of equivalence classes of ways of placing 2 3 X 3 square tiles in a 6 X 6 square under all symmetry operations of the square is 6. The portrayal of an example from each equivalence class is:
%e A236560 .___________      ___________      ___________
%e A236560 |     |     |    |     |_____|    |     |     |
%e A236560 |  .  |  .  |    |  .  |     |    |  .  |_____|
%e A236560 |_____|_____|    |_____|  .  |    |_____|     |
%e A236560 |           |    |     |_____|    |     |  .  |
%e A236560 |           |    |           |    |     |_____|
%e A236560 |___________|    |___________|    |_____|_____|
%e A236560 .
%e A236560 .___________      ___________      ___________
%e A236560 |     |     |    |_____ _____|    |_____      |
%e A236560 |  .  |     |    |     |     |    |     |_____|
%e A236560 |_____|_____|    |  .  |  .  |    |  .  |     |
%e A236560 |     |     |    |_____|_____|    |_____|  .  |
%e A236560 |     |  .  |    |           |    |     |_____|
%e A236560 |_____|_____|    |___________|    |_____|_____|
%Y A236560 Cf. A054252, A236679, A236757, A236800, A236829, A236865, A236915, A236936, A236939.
%K A236560 tabf,nonn
%O A236560 3,6
%A A236560 _Christopher Hunt Gribble_, Jan 30 2014