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A236865 Number T(n,k) of equivalence classes of ways of placing k 7 X 7 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=7, 0<=k<=floor(n/7)^2, read by rows. 9
1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 1, 10, 20, 4, 1, 1, 15, 65, 59, 14, 1, 15, 153, 329, 174, 1, 21, 295, 1225, 1234, 1, 21, 507, 3465, 6124, 1, 28, 810, 8358, 23259, 1, 28, 1214, 17710, 73204 (list; graph; refs; listen; history; text; internal format)
OFFSET

7,6

COMMENTS

The first 13 rows of T(n,k) are:

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

n

7     1     1

8     1     1

9     1     3

10    1     3

11    1     6

12    1     6

13    1    10

14    1    10    20     4     1

15    1    15    65    59    14

16    1    15   153   329   174

17    1    21   295  1225  1234

18    1    21   507  3465  6124

19    1    28   810  8358 23259

20    1    28  1214 17710 73204

LINKS

Table of n, a(n) for n=7..55.

Christopher Hunt Gribble, C++ program

FORMULA

It appears that:

T(n,0) = 1, n>= 7

T(n,1) = (floor((n-7)/2)+1)*(floor((n-7)/2+2))/2, n >= 7

T(c+2*7,2) = A131474(c+1)*(7-1) + A000217(c+1)*floor(7^2/4) + A014409(c+2), 0 <= c < 7, c even

T(c+2*7,2) = A131474(c+1)*(7-1) + A000217(c+1)*floor((7-1)(7-3)/4) + A014409(c+2), 0 <= c < 7, c odd

T(c+2*7,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((7-c-1)/2) + A131941(c+1)*floor((7-c)/2)) + S(c+1,3c+2,3), 0 <= c < 7 where

S(c+1,3c+2,3) =

A054252(2,3),  c = 0

A236679(5,3),  c = 1

A236560(8,3),  c = 2

A236757(11,3), c = 3

A236800(14,3), c = 4

A236829(17,3), c = 5

A236865(20,3), c = 6

EXAMPLE

T(14,3) = 4 because the number of equivalent classes of ways of placing 3 7 X 7 square tiles in an 14 X 14 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:

.___________________          ___________________

|         |         |        |         |_________|

|         |         |        |         |         |

|         |         |        |         |         |

|    .    |    .    |        |    .    |         |

|         |         |        |         |    .    |

|         |         |        |         |         |

|_________|_________|        |_________|         |

|         |         |        |         |_________|

|         |         |        |         |         |

|         |         |        |         |         |

|    .    |         |        |    .    |         |

|         |         |        |         |         |

|         |         |        |         |         |

|_________|_________|        |_________|_________|

.

.___________________          ___________________

|         |         |        |         |         |

|         |_________|        |         |         |

|         |         |        |         |_________|

|    .    |         |        |    .    |         |

|         |         |        |         |         |

|         |    .    |        |         |         |

|_________|         |        |_________|    .    |

|         |         |        |         |         |

|         |_________|        |         |         |

|         |         |        |         |_________|

|    .    |         |        |    .    |         |

|         |         |        |         |         |

|         |         |        |         |         |

|_________|_________|        |_________|_________|

CROSSREFS

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236915, A236936, A236939.

Sequence in context: A236939 A236936 A236915 * A236829 A236800 A126212

Adjacent sequences:  A236862 A236863 A236864 * A236866 A236867 A236868

KEYWORD

tabf,nonn

AUTHOR

Christopher Hunt Gribble, Jan 31 2014

STATUS

approved

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Last modified June 24 21:51 EDT 2022. Contains 354830 sequences. (Running on oeis4.)