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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.
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%I #25 Mar 18 2023 08:37:31

%S 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,

%T 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,

%U 115,39,21,1,1,1,1,30,82,295,521,521,295,82,30,1,1

%N 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.

%H Christopher Hunt Gribble, <a href="/A227690/b227690.txt">Antidiagonals n = 0..15, flattened</a>

%H Christopher Hunt Gribble, <a href="/A227690/a227690.cpp.txt">C++ program</a>

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 2, 4, 5, 9, 12, 21, ...

%e 1, 1, 2, 3, 6, 10, 21, 39, 82, ...

%e 1, 1, 4, 6, 13, 39, 115, 295, 861, ...

%e 1, 1, 5, 10, 39, 77, 521, 1985, 8038, ...

%e 1, 1, 9, 21, 115, 521, 1494, 15129, 83609, ...

%e 1, 1, 12, 39, 295, 1985, 15129, 56978, 861159, ...

%e 1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023, ...

%e ...

%e 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:

%e ._____ _. ._______. ._______.

%e | |_| | | | | |_|_|

%e | |_| |___|_ _| |___| |

%e |_____|_| |_|_|_|_| |_|_|___|

%e ._______. ._______. ._______.

%e | |_|_| |_| |_| |_|_|_|_|

%e |___|_|_| |_|___|_| |_|_|_|_|

%e |_|_|_|_| |_|_|_|_| |_|_|_|_|

%Y Main diagonal: A224239.

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

%Y Cf. A219924, A224697.

%K nonn,tabl

%O 0,13

%A _Christopher Hunt Gribble_, Jul 19 2013