A225910 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.

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

Offset: 0

Author

Yosu Yurramendi, May 20 2013

Keywords

• nonn
• easy
• tabl

Comments

In the square table A000012, A005418, and A225826 to A225834 are the first 11 rows (see example).

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)).

All the coefficients of x in generating functions from A225826 to A225834 belong to A113979.

Examples

			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 ...
  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..65, flattened
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Crossrefs

Cf. A005418, A225826-A225834.

Formula

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.