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%I #27 Nov 23 2023 08:02:09
%S 1,1,1,1,2,1,1,3,3,1,1,6,7,6,1,1,10,24,24,10,1,1,20,76,168,76,20,1,1,
%T 36,288,1120,1120,288,36,1,1,72,1072,8640,16576,8640,1072,72,1,1,136,
%U 4224,66816,263680,263680,66816,4224,136,1,1,272,16576,529920,4197376,8407040,4197376,529920,16576,272,1
%N Square array read by antidiagonals: a(m,n) is the number of binary pattern classes in the (m,n)-rectangular grid, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
%C In the square table A000012, A005418, and A225826 to A225834 are the first 11 rows (see example).
%C In the square table, m odd (see formula). The order of the recurrence equations is 4. Let it be (a1(m),a2(m),a3(m),a4(m)) the characterizing 4-plet of a(m). The sequence a1(m) belongs to A028403 (2^m+2^((m+1)/2)), -a2(m) to A147538 (2^m*(2^((m+1)/2)-1) and a4(m) to A013824 (2^(2m)*2^((m+1)/2)). -a3(m) sequence formula is 2^m*(2^m+2^((m+1)/2)).
%C All the coefficients of x in generating functions from A225826 to A225834 belong to A113979.
%H Alois P. Heinz, <a href="/A225910/b225910.txt">Antidiagonals n = 0..65, flattened</a>
%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv:2311.13072 [math.CO], 2023.
%F m even and n even:
%F a(m,n) = 2^(m*n/2-2)*(2^(m*n/2) + 3);
%F m even and n odd:
%F a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(m/2-1) + 1);
%F m odd and n even:
%F a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(n/2-1) + 1);
%F m odd and n odd:
%F a(m,n) = 2^((m*n-1)/2-1)*(2^((m*n-1)/2) + 2^((m-1)/2) + 2^((n-1)/2) + 1).
%F m even:
%F a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
%F m odd:
%F a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) - 2^(((m+1)/2)*n-3)*(2^((m-1)/2)-1) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
%F Only a(1,n) and a(2,n) (A005418 and A225826) sequences are needed to define the others.
%e Array begins:
%e 1 1 1 1 1 1 1 ...
%e 1 2 3 6 10 20 36 ...
%e 1 3 7 24 76 288 1072 ...
%e 1 6 24 168 1120 8640 66816 ...
%e 1 10 76 1120 16576 263680 4197376 ...
%e 1 20 288 8640 263680 8407040 268517376 ...
%e 1 36 1072 66816 4197376 268517376 17180065792 ...
%e 1 72 4224 529920 67133440 8590786560 1099516870656 ...
%e 1 136 16576 4212736 1073790976 274882625536 70368756760576 ...
%e 1 272 66048 33632256 17180262400 8796137062400 4503599962914816 ...
%e 1 528 262912 268713984 274878693376 281475261923328 288230376957018112 ...
%e ...
%Y Cf. A005418, A225826-A225834.
%K nonn,easy,tabl
%O 0,5
%A _Yosu Yurramendi_, May 20 2013
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