A236915 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.

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, 377, 174, 1, 21, 351, 1365, 1234, 1, 28, 606, 3900, 6124, 1, 28, 948, 9282, 23259, 1, 36, 1426, 19726, 73204, 1, 36, 2026, 38046, 199436

Offset

8,6

Comments

The first 16 rows of T(n,k) are:

.\ k 0 1 2 3 4

n

8 1 1

9 1 1

10 1 3

11 1 3

12 1 6

13 1 6

14 1 10

15 1 10

16 1 15 25 5 1

17 1 15 79 65 14

18 1 21 187 377 174

19 1 21 351 1365 1234

20 1 28 606 3900 6124

21 1 28 948 9282 23259

22 1 36 1426 19726 73204

23 1 36 2026 38046 199436

Links

Christopher Hunt Gribble, Rows n = 8..23, flattened

Christopher Hunt Gribble, C++ program

Formula

It appears that:

T(n,0) = 1, n>= 8

T(n,1) = (floor((n-8)/2)+1)*(floor((n-8)/2+2))/2, n >= 8

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

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

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

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

A236915(23,3), c = 7

Example

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:
._____________________        _____________________
|          |          |      |          |__________|
|          |          |      |          |          |
|          |          |      |          |          |
|     .    |     .    |      |     .    |          |
|          |          |      |          |     .    |
|          |          |      |          |          |
|          |          |      |          |          |
|__________|__________|      |__________|          |
|          |          |      |          |__________|
|          |          |      |          |          |
|          |          |      |          |          |
|    .     |          |      |     .    |          |
|          |          |      |          |          |
|          |          |      |          |          |
|          |          |      |          |          |
|__________|__________|      |__________|__________|
.
._____________________        _____________________
|          |          |      |          |          |
|          |__________|      |          |          |
|          |          |      |          |__________|
|     .    |          |      |     .    |          |
|          |          |      |          |          |
|          |     .    |      |          |          |
|          |          |      |          |     .    |
|__________|          |      |__________|          |
|          |          |      |          |          |
|          |__________|      |          |          |
|          |          |      |          |__________|
|     .    |          |      |     .    |          |
|          |          |      |          |          |
|          |          |      |          |          |
|          |          |      |          |          |
|__________|__________|      |__________|__________|
.
._____________________
|          |          |
|          |          |
|          |          |
|     .    |__________|
|          |          |
|          |          |
|          |          |
|__________|     .    |
|          |          |
|          |          |
|          |          |
|     .    |__________|
|          |          |
|          |          |
|          |          |
|__________|__________|

Crossrefs

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236936, A236939.

Sequence in context: A320221 A236939 A236936 * A236865 A236829 A236800

Adjacent sequences: A236912 A236913 A236914 * A236916 A236917 A236918

Keyword

tabf,nonn

Author

Christopher Hunt Gribble, Feb 01 2014

Status

approved