A236829 Number T(n,k) of equivalence classes of ways of placing k 6 X 6 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=6, 0<=k<=floor(n/6)^2, read by rows.

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 16, 4, 1, 1, 10, 51, 50, 14, 1, 15, 125, 293, 174, 1, 15, 239, 1065, 1234, 1, 21, 423, 3075, 6124, 1, 21, 672, 7371, 23259, 1, 28, 1030, 16093, 81480, 51615, 10596, 808, 31, 1

Offset

6,6

Comments

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

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

n

6 1 1

7 1 1

8 1 3

9 1 3

10 1 6

11 1 6

12 1 10 16 4 1

13 1 10 51 50 14

14 1 15 125 293 174

15 1 15 239 1065 1234

16 1 21 423 3075 6124

17 1 21 672 7371 23259

18 1 28 1030 16093 81480 51615 10596 808 31 1

Links

Table of n, a(n) for n=6..57.

Christopher Hunt Gribble, C++ program

Formula

It appears that:

T(n,0) = 1, n>= 6

T(n,1) = (floor((n-6)/2)+1)*(floor((n-6)/2+2))/2, n >= 6

T(c+2*6,2) = A131474(c+1)*(6-1) + A000217(c+1)*floor(6^2/4) + A014409(c+2), 0 <= c < 6, c even

T(c+2*6,2) = A131474(c+1)*(6-1) + A000217(c+1)*floor((6-1)(6-3)/4) + A014409(c+2), 0 <= c < 6, c odd

T(c+2*6,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((6-c-1)/2) + A131941(c+1)*floor((6-c)/2)) + S(c+1,3c+2,3), 0 <= c < 6 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

Example

T(12,3) = 4 because the number of equivalence classes of ways of placing 3 6 X 6 square tiles in a 12 X 12 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, A236865, A236915, A236936, A236939.

Sequence in context: A236936 A236915 A236865 * A236800 A367628 A126212

Adjacent sequences: A236826 A236827 A236828 * A236830 A236831 A236832

Keyword

tabf,nonn

Author

Christopher Hunt Gribble, Jan 31 2014

Status

approved