login
Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5)^2, read by rows.
9

%I #21 Feb 18 2014 13:30:41

%S 1,1,1,1,1,3,1,3,1,6,1,6,12,3,1,1,10,40,44,14,1,10,97,245,174,1,15,

%T 193,925,1234,1,15,339,2640,6124,1,21,555,6617,27074,19336,4785,461,

%U 23,1

%N Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5)^2, read by rows.

%C The first 11 rows of T(n,k) are:

%C .\ k 0 1 2 3 4 5 6 7 8 9

%C n

%C 5 1 1

%C 6 1 1

%C 7 1 3

%C 8 1 3

%C 9 1 6

%C 10 1 6 12 3 1

%C 11 1 10 40 44 14

%C 12 1 10 97 245 174

%C 13 1 15 193 925 1234

%C 14 1 15 339 2640 6124

%C 15 1 21 555 6617 27074 19336 4785 461 23 1

%H Christopher Hunt Gribble, <a href="/A236800/a236800.cpp.txt">C++ program</a>

%F It appears that:

%F T(n,0) = 1, n>= 5

%F T(n,1) = (floor((n-5)/2)+1)*(floor((n-5)/2+2))/2, n >= 5

%F T(c+2*5,2) = A131474(c+1)*(5-1) + A000217(c+1)*floor(5^2/4) + A014409(c+2), 0 <= c < 5, c even

%F T(c+2*5,2) = A131474(c+1)*(5-1) + A000217(c+1)*floor((5-1)(5-3)/4) + A014409(c+2), 0 <= c < 5, c odd

%F T(c+2*5,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((5-c-1)/2) + A131941(c+1)*floor((5-c)/2)) + S(c+1,3c+2,3), 0 <= c < 5 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

%e T(10,3) = 3 because the number of equivalence classes of ways of placing 3 5 X 5 square tiles in an 10 X 10 square under all symmetry operations of the square is 3. The portrayal of an example from each equivalence class is:

%e ._______________ _______________ _______________

%e | | | | |_______| | | |

%e | | | | | | | |_______|

%e | . | . | | . | | | . | |

%e | | | | | . | | | |

%e |_______|_______| |_______| | |_______| . |

%e | | | | |_______| | | |

%e | | | | | | | |_______|

%e | . | | | . | | | . | |

%e | | | | | | | | |

%e |_______|_______| |_______|_______| |_______|_______|

%Y Cf. A054252, A236679, A236560, A236757, A236829, A236865, A236915, A236936, A236939.

%K tabf,nonn

%O 5,6

%A _Christopher Hunt Gribble_, Jan 31 2014