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.

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

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