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A243866 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing one 1 X 1 tile in an n X k rectangle under all symmetry operations of the rectangle. 6
1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 3, 3, 4, 4, 3, 3, 4, 3, 6, 4, 6, 3, 4, 4, 4, 6, 6, 6, 6, 4, 4, 5, 4, 8, 6, 9, 6, 8, 4, 5, 5, 5, 8, 8, 9, 9, 8, 8, 5, 5, 6, 5, 10, 8, 12, 9, 12, 8, 10, 5, 6, 6, 6, 10, 10, 12, 12, 12, 12, 10, 10, 6, 6, 7, 6, 12, 10, 15 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

It appears that the number of equivalence classes of ways of placing one m X m tile in an n X k rectangle under all symmetry operations of the rectangle is T(n-m+1,k-m+1) for m >= 2, n >= m and k >= m, and zero otherwise. - Christopher Hunt Gribble, Oct 17 2014

The sum over each antidiagonal of A243866

= Sum_{j=1..n}(2*j + 1 - (-1)^j)*(2*(n - j + 1) + 1 - (-1)^(n - j + 1))/16

= (n + 2)*(2*n^2 + 8*n + 3 - 3*(-1)^n)/48

- see A006918. - Christopher Hunt Gribble, Apr 01 2015

LINKS

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

FORMULA

Empirically,

T(n,k) = floor((n+1)/2)*floor((k+1)/2)

       = (2*n+1-(-1)^n)*(2*k+1-(-1)^k)/4;

T(n,1) = A034851(n,1);

T(n,2) = A226048(n,1);

T(n,3) = A226290(n,1);

T(n,4) = A225812(n,1);

T(n,5) = A228022(n,1);

T(n,6) = A228165(n,1);

T(n,7) = A228166(n,1). - Christopher Hunt Gribble, May 01 2015

EXAMPLE

T(n,k) for 1<=n<=11 and 1<=k<=11 is:

    k    1    2    3    4    5    6    7    8    9   10   11 ...

.n

.1       1    1    2    2    3    3    4    4    5    5    6

.2       1    1    2    2    3    3    4    4    5    5    6

.3       2    2    4    4    6    6    8    8   10   10   12

.4       2    2    4    4    6    6    8    8   10   10   12

.5       3    3    6    6    9    9   12   12   15   15   18

.6       3    3    6    6    9    9   12   12   15   15   18

.7       4    4    8    8   12   12   16   16   20   20   24

.8       4    4    8    8   12   12   16   16   20   20   24

.9       5    5   10   10   15   15   20   20   25   25   30

10       5    5   10   10   15   15   20   20   25   25   30

11       6    6   12   12   18   18   24   24   30   30   36

...

MAPLE

b := proc (n, k);

floor((1/2)*n+1/2)*floor((1/2)*k+1/2)

end proc;

seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);

CROSSREFS

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248017, A248027.

Sequence in context: A339930 A307707 A025819 * A110102 A255472 A230198

Adjacent sequences:  A243863 A243864 A243865 * A243867 A243868 A243869

KEYWORD

tabl,nonn

AUTHOR

Christopher Hunt Gribble, Jun 19 2014

EXTENSIONS

Terms corrected by Christopher Hunt Gribble, Mar 27 2015

STATUS

approved

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Last modified October 21 11:09 EDT 2021. Contains 348150 sequences. (Running on oeis4.)