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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.

%I A227690 #25 Mar 18 2023 08:37:31
%S A227690 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 A227690 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 A227690 115,39,21,1,1,1,1,30,82,295,521,521,295,82,30,1,1
%N 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.
%H A227690 Christopher Hunt Gribble, <a href="/A227690/b227690.txt">Antidiagonals n = 0..15, flattened</a>
%H A227690 Christopher Hunt Gribble, <a href="/A227690/a227690.cpp.txt">C++ program</a>
%e A227690 Square array A(n,k) begins:
%e A227690   1, 1,  1,  1,   1,    1,     1,      1,       1, ...
%e A227690   1, 1,  1,  1,   1,    1,     1,      1,       1, ...
%e A227690   1, 1,  2,  2,   4,    5,     9,     12,      21, ...
%e A227690   1, 1,  2,  3,   6,   10,    21,     39,      82, ...
%e A227690   1, 1,  4,  6,  13,   39,   115,    295,     861, ...
%e A227690   1, 1,  5, 10,  39,   77,   521,   1985,    8038, ...
%e A227690   1, 1,  9, 21, 115,  521,  1494,  15129,   83609, ...
%e A227690   1, 1, 12, 39, 295, 1985, 15129,  56978,  861159, ...
%e A227690   1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023, ...
%e A227690 ...
%e A227690 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 A227690    ._____ _.    ._______.    ._______.
%e A227690    |     |_|    |   |   |    |   |_|_|
%e A227690    |     |_|    |___|_ _|    |___|   |
%e A227690    |_____|_|    |_|_|_|_|    |_|_|___|
%e A227690    ._______.    ._______.    ._______.
%e A227690    |   |_|_|    |_|   |_|    |_|_|_|_|
%e A227690    |___|_|_|    |_|___|_|    |_|_|_|_|
%e A227690    |_|_|_|_|    |_|_|_|_|    |_|_|_|_|
%Y A227690 Main diagonal: A224239.
%Y A227690 Columns 1-10: A000012, A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.
%Y A227690 Cf. A219924, A224697.
%K A227690 nonn,tabl
%O A227690 0,13
%A A227690 _Christopher Hunt Gribble_, Jul 19 2013