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A225910
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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.
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22
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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, 36, 288, 1120, 1120, 288, 36, 1, 1, 72, 1072, 8640, 16576, 8640, 1072, 72, 1, 1, 136, 4224, 66816, 263680, 263680, 66816, 4224, 136, 1, 1, 272, 16576, 529920, 4197376, 8407040, 4197376, 529920, 16576, 272, 1
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OFFSET
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0,5
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COMMENTS
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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)).
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LINKS
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FORMULA
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m even and n even:
a(m,n) = 2^(m*n/2-2)*(2^(m*n/2) + 3);
m even and n odd:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(m/2-1) + 1);
m odd and n even:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(n/2-1) + 1);
m odd and n odd:
a(m,n) = 2^((m*n-1)/2-1)*(2^((m*n-1)/2) + 2^((m-1)/2) + 2^((n-1)/2) + 1).
m even:
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).
m odd:
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).
Only a(1,n) and a(2,n) (A005418 and A225826) sequences are needed to define the others.
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EXAMPLE
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Array begins:
1 1 1 1 1 1 1 ...
1 2 3 6 10 20 36 ...
1 3 7 24 76 288 1072 ...
1 6 24 168 1120 8640 66816 ...
1 10 76 1120 16576 263680 4197376 ...
1 20 288 8640 263680 8407040 268517376 ...
1 36 1072 66816 4197376 268517376 17180065792 ...
1 72 4224 529920 67133440 8590786560 1099516870656 ...
1 136 16576 4212736 1073790976 274882625536 70368756760576 ...
1 272 66048 33632256 17180262400 8796137062400 4503599962914816 ...
1 528 262912 268713984 274878693376 281475261923328 288230376957018112 ...
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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