A286919 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 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, 8, 1, 36, 1072, 1, 288, 66816, 33693696, 1, 2080, 4197376, 17184194560, 70368756760576, 1, 16640, 268517376, 8796399206400, 288230393868451840, 9444732983468915425280, 1, 131328, 17180065792, 4503616874348544, 1180591620768950910976, 309485009825866260538195968, 81129638414606695206587887255552

Offset: 0

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Keywords

• nonn
• tabl

Comments

Computed using Burnsides orbit-counting lemma.

Examples

			Triangle begins:
========================================================
n\m |   0   1      2        3             4
----|---------------------------------------------------
0   |   1
1   |   1   8
2   |   1   36     1072
3   |   1   288    66816    33693696
4   |   1   2080   4197376  17184194560   70368756760576
...
		

Links

• María Merino, Rows n=0..35 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, A283433, A283434, A286893, A286895.

Formula

For even n and m: T(n,m) = (8^(m*n) + 3*8^(m*n/2))/4;

for even n and odd m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 2*8^(m*n/2))/4;

for odd n and even m: T(n,m) = (8^(m*n) + 8^((m*n+m)/2) + 2*8^(m*n/2))/4;

for odd n and m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 8^((m*n+m)/2) + 8^((m*n+1)/2))/4.