A227690 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles reduced for symmetry; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 6, 6, 5, 1, 1, 1, 1, 9, 10, 13, 10, 9, 1, 1, 1, 1, 12, 21, 39, 39, 21, 12, 1, 1, 1, 1, 21, 39, 115, 77, 115, 39, 21, 1, 1, 1, 1, 30, 82, 295, 521, 521, 295, 82, 30, 1, 1

Offset

0,13

Links

Christopher Hunt Gribble, Antidiagonals n = 0..15, flattened

Christopher Hunt Gribble, C++ program

Example

Square array A(n,k) begins:
  1, 1,  1,  1,   1,    1,     1,      1,       1, ...
  1, 1,  1,  1,   1,    1,     1,      1,       1, ...
  1, 1,  2,  2,   4,    5,     9,     12,      21, ...
  1, 1,  2,  3,   6,   10,    21,     39,      82, ...
  1, 1,  4,  6,  13,   39,   115,    295,     861, ...
  1, 1,  5, 10,  39,   77,   521,   1985,    8038, ...
  1, 1,  9, 21, 115,  521,  1494,  15129,   83609, ...
  1, 1, 12, 39, 295, 1985, 15129,  56978,  861159, ...
  1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023, ...
...
A(4,3) = 6 because there are 6 ways to tile a 3 X 4 rectangle by subsquares, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct:
   ._____ _.    ._______.    ._______.
   |     |_|    |   |   |    |   |_|_|
   |     |_|    |___|_ _|    |___|   |
   |_____|_|    |_|_|_|_|    |_|_|___|
   ._______.    ._______.    ._______.
   |   |_|_|    |_|   |_|    |_|_|_|_|
   |___|_|_|    |_|___|_|    |_|_|_|_|
   |_|_|_|_|    |_|_|_|_|    |_|_|_|_|

Crossrefs

Main diagonal: A224239.

Columns 1-10: A000012, A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.

Cf. A219924, A224697.

Sequence in context: A049704 A047996 A377060 * A063686 A333159 A008327

Adjacent sequences: A227687 A227688 A227689 * A227691 A227692 A227693

Keyword

nonn,tabl

Author

Christopher Hunt Gribble, Jul 19 2013

Status

approved