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A361001 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling), 1 <= k <= n.

%I A361001 #7 Mar 02 2023 06:22:18
%S A361001 1,2,4,3,7,9,4,11,18,23,4,14,22,34,41,6,23,42,72,108
%N A361001 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling), 1 <= k <= n.
%F A361001 T(n,1) = A361003(n) = A000005(n) + floor((n-1)/2). (The first term corresponds to cases where all pieces have the same size, the second to cases where there are two pieces of different sizes.)
%e A361001 Triangle begins:
%e A361001   n\k|  1  2  3  4   5  6
%e A361001   ---+-------------------
%e A361001   1  |  1
%e A361001   2  |  2  4
%e A361001   3  |  3  7  9
%e A361001   4  |  4 11 18 23
%e A361001   5  |  4 14 22 34  41
%e A361001   6  |  6 23 42 72 108  ?
%e A361001 The T(3,3) = 9 nonrearrangeable tilings of the 3 X 3 square are:
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%e A361001   |           |   |           |   |       |   |
%e A361001   +           +   +---+---+---+   +---+---+---+
%e A361001   |           |   |           |   |           |
%e A361001   +           +   +           +   +           +
%e A361001   |           |   |           |   |           |
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%e A361001 .
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%e A361001   |   |   |   |   |       |   |   |   |   |   |
%e A361001   +---+---+---+   +---+---+   +   +---+---+   +
%e A361001   |           |   |       |   |   |       |   |
%e A361001   +           +   +       +   +   +       +   +
%e A361001   |           |   |       |   |   |       |   |
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%e A361001 .
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%e A361001   |   |   |   |   |           |   |   |   |   |
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%e A361001   |       |   |   |           |   |   |   |   |
%e A361001   +       +---+   +---+---+---+   +---+---+---+
%e A361001   |       |   |   |           |   |   |   |   |
%e A361001   +---+---+---+   +---+---+---+   +---+---+---+
%Y A361001 Cf. A000005, A360629, A360998, A361002 (main diagonal), A361003 (first column), A361004 (second column), A361005 (third column).
%K A361001 nonn,tabl,more
%O A361001 1,2
%A A361001 _Pontus von Brömssen_, Feb 28 2023