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%I #30 Feb 17 2014 12:55:22
%S 1,1,1,1,1,3,1,3,1,6,1,6,1,10,1,10,1,15,25,5,1,1,15,79,65,14,1,21,187,
%T 377,174,1,21,351,1365,1234,1,28,606,3900,6124,1,28,948,9282,23259,1,
%U 36,1426,19726,73204,1,36,2026,38046,199436
%N Number T(n,k) of equivalence classes of ways of placing k 8 X 8 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=8, 0<=k<=floor(n/8)^2, read by rows.
%C The first 16 rows of T(n,k) are:
%C .\ k 0 1 2 3 4
%C n
%C 8 1 1
%C 9 1 1
%C 10 1 3
%C 11 1 3
%C 12 1 6
%C 13 1 6
%C 14 1 10
%C 15 1 10
%C 16 1 15 25 5 1
%C 17 1 15 79 65 14
%C 18 1 21 187 377 174
%C 19 1 21 351 1365 1234
%C 20 1 28 606 3900 6124
%C 21 1 28 948 9282 23259
%C 22 1 36 1426 19726 73204
%C 23 1 36 2026 38046 199436
%H Christopher Hunt Gribble, <a href="/A236915/b236915.txt">Rows n = 8..23, flattened</a>
%H Christopher Hunt Gribble, <a href="/A236915/a236915.cpp.txt">C++ program</a>
%F It appears that:
%F T(n,0) = 1, n>= 8
%F T(n,1) = (floor((n-8)/2)+1)*(floor((n-8)/2+2))/2, n >= 8
%F T(c+2*8,2) = A131474(c+1)*(8-1) + A000217(c+1)*floor(8^2/4) + A014409(c+2), 0 <= c < 8, c even
%F T(c+2*8,2) = A131474(c+1)*(8-1) + A000217(c+1)*floor((8-1)(8-3)/4) + A014409(c+2), 0 <= c < 8, c odd
%F T(c+2*8,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((8-c-1)/2) + A131941(c+1)*floor((8-c)/2)) + S(c+1,3c+2,3), 0 <= c < 8 where
%F S(c+1,3c+2,3) =
%F A054252(2,3), c = 0
%F A236679(5,3), c = 1
%F A236560(8,3), c = 2
%F A236757(11,3), c = 3
%F A236800(14,3), c = 4
%F A236829(17,3), c = 5
%F A236865(20,3), c = 6
%F A236915(23,3), c = 7
%e T(16,3) = 5 because the number of equivalence classes of ways of placing 3 8 X 8 square tiles in an 16 X 16 square under all symmetry operations of the square is 5. The portrayal of an example from each equivalence class is:
%e ._____________________ _____________________
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%e |__________|__________| |__________|__________|
%e .
%e ._____________________ _____________________
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%e |__________|__________| |__________|__________|
%e .
%e ._____________________
%e | | |
%e | | |
%e | | |
%e | . |__________|
%e | | |
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%e |__________| . |
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%Y Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236936, A236939.
%K tabf,nonn
%O 8,6
%A _Christopher Hunt Gribble_, Feb 01 2014