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A236560 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3)^2, read by rows. 9
1, 1, 1, 1, 1, 3, 1, 3, 6, 2, 1, 1, 6, 21, 29, 14, 1, 6, 53, 161, 174, 1, 10, 111, 665, 1713, 1549, 608, 107, 11, 1, 1, 10, 201, 1961, 9973, 24267, 29437, 17438, 4756, 459 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,6

COMMENTS

The first 8 rows of T(n,k) are:

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

n

3     1     1

4     1     1

5     1     3

6     1     3     6     2     1

7     1     6    21    29    14

8     1     6    53   161   174

9     1    10   111   665  1713  1549   608   107    11     1

10    1    10   201  1961  9973 24267 29437 17438  4756   459

LINKS

Table of n, a(n) for n=3..43.

Christopher Hunt Gribble, C++ program

FORMULA

It appears that:

T(n,0) = 1, n>= 3

T(n,1) = (floor((n-3)/2)+1)*(floor((n-3)/2+2))/2, n >= 3

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

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

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

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

A054252(2,3),  c = 0

A236679(5,3),  c = 1

A236560(8,3),  c = 2

EXAMPLE

T(6,2) = 6 because the number of equivalence classes of ways of placing 2 3 X 3 square tiles in a 6 X 6 square under all symmetry operations of the square is 6. The portrayal of an example from each equivalence class is:

.___________      ___________      ___________

|     |     |    |     |_____|    |     |     |

|  .  |  .  |    |  .  |     |    |  .  |_____|

|_____|_____|    |_____|  .  |    |_____|     |

|           |    |     |_____|    |     |  .  |

|           |    |           |    |     |_____|

|___________|    |___________|    |_____|_____|

.

.___________      ___________      ___________

|     |     |    |_____ _____|    |_____      |

|  .  |     |    |     |     |    |     |_____|

|_____|_____|    |  .  |  .  |    |  .  |     |

|     |     |    |_____|_____|    |_____|  .  |

|     |  .  |    |           |    |     |_____|

|_____|_____|    |___________|    |_____|_____|

CROSSREFS

Cf. A054252, A236679, A236757, A236800, A236829, A236865, A236915, A236936, A236939.

Sequence in context: A096713 A107726 A114159 * A291723 A255974 A033789

Adjacent sequences:  A236557 A236558 A236559 * A236561 A236562 A236563

KEYWORD

tabf,nonn

AUTHOR

Christopher Hunt Gribble, Jan 30 2014

STATUS

approved

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Last modified November 27 12:51 EST 2021. Contains 349394 sequences. (Running on oeis4.)