A283433 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 4 colors and n>=m, 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, 4, 1, 10, 76, 1, 40, 1120, 67840, 1, 136, 16576, 4212736, 1073790976, 1, 544, 263680, 268779520, 274882625536, 281475530358784, 1, 2080, 4197376, 17184194560, 70368756760576, 288230393868451840, 1180591620768950910976

Offset: 0

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Keywords

• nonn
• tabl

Comments

Computed using Burnside's orbit-counting lemma.

Examples

			Triangle begins:
=======================================================================
n\m |  0   1       2        3           4               5
----|------------------------------------------------------------------
0   |  1
1   |  1    4
2   |  1    10     76
3   |  1    40     1120     67840
4   |  1    136    16576    4212736     1073790976
5   |  1    544    263680   268779520   274882625536    281475530358784
...
		

Links

• María Merino, Rows n=0..41 of triangle, flattened
• M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Crossrefs

Cf. A225910, A283432.

Formula

For even n and m: T(n,m) = (4^(m*n) + 3*4^(m*n/2))/4;

for even n and odd m: T(n,m) = (4^(m*n) + 4^((m*n+n)/2) + 2*4^(m*n/2))/4;

for odd n and even m: T(n,m) = (4^(m*n) + 4^((m*n+m)/2) + 2*4^(m*n/2))/4;

for odd n and m: T(n,m) = (4^(m*n) + 4^((m*n+n)/2) + 4^((m*n+m)/2) + 4^((m*n+1)/2))/4.