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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
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
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:
.___________ ___________ ___________
| | | | |_____| | | |
| . | . | | . | | | . |_____|
|_____|_____| |_____| . | |_____| |
| | | |_____| | | . |
| | | | | |_____|
|___________| |___________| |_____|_____|
.
.___________ ___________ ___________
| | | |_____ _____| |_____ |
| . | | | | | | |_____|
|_____|_____| | . | . | | . | |
| | | |_____|_____| |_____| . |
| | . | | | | |_____|
|_____|_____| |___________| |_____|_____|
KEYWORD
tabf,nonn
AUTHOR
STATUS
approved