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Number T(n,k) of equivalence classes of ways of placing k 9 X 9 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=9, 0<=k<=floor(n/9)^2, read by rows.
9

%I #21 Feb 17 2014 12:59:17

%S 1,1,1,1,1,3,1,3,1,6,1,6,1,10,1,10,1,15,1,15,30,5,1,1,21,96,74,14,1,

%T 21,221,413,174,1,28,417,1525,1234,1,28,705,4290,6124,1,36,1107,10269,

%U 23259,1,36,1638,21630,73204,1,45,2334,41790,199436

%N Number T(n,k) of equivalence classes of ways of placing k 9 X 9 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=9, 0<=k<=floor(n/9)^2, read by rows.

%H Christopher Hunt Gribble, <a href="/A236936/a236936.cpp.txt">C++ program</a>

%H Christopher Hunt Gribble, <a href="/A236936/a236936.txt">Example graphics</a>

%F It appears that:

%F T(n,0) = 1, n>= 9

%F T(n,1) = (floor((n-9)/2)+1)*(floor((n-9)/2+2))/2, n >= 9

%F T(c+2*9,2) = A131474(c+1)*(9-1) + A000217(c+1)*floor(9^2/4) + A014409(c+2), 0 <= c < 9, c even

%F T(c+2*9,2) = A131474(c+1)*(9-1) + A000217(c+1)*floor((9-1)(9-3)/4) + A014409(c+2), 0 <= c < 9, c odd

%F T(c+2*9,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((9-c-1)/2) + A131941(c+1)*floor((9-c)/2)) + S(c+1,3c+2,3), 0 <= c < 9 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

%F A236936(26,3), c = 8

%e The first 17 rows of T(n,k) are:

%e .\ k 0 1 2 3 4

%e n

%e 9 1 1

%e 10 1 1

%e 11 1 3

%e 12 1 3

%e 13 1 6

%e 14 1 6

%e 15 1 10

%e 16 1 10

%e 17 1 15

%e 18 1 15 30 5 1

%e 19 1 21 96 74 14

%e 20 1 21 221 413 174

%e 21 1 28 417 1525 1234

%e 22 1 28 705 4290 6124

%e 23 1 36 1107 10269 23259

%e 24 1 36 1638 21630 73204

%e 25 1 45 2334 41790 199436

%e .

%e T(18,3) = 5 because the number of equivalence classes of ways of placing 3 9 X 9 square tiles in an 18 X 18 square under all symmetry operations of the square is 5.

%Y Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236915, A236939.

%K tabf,nonn

%O 9,6

%A _Christopher Hunt Gribble_, Feb 01 2014