A286920 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 9 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, 9, 1, 45, 1701, 1, 405, 134865, 97135605, 1, 3321, 10766601, 70618411521, 463255079498001, 1, 29889, 871858485, 51473762336565, 3039416437115008521, 179474497026544179696969, 1, 266085, 70607782701, 37523729625344145, 19941610769429949618201, 10597789568841677482963905405, 5632099886234793715531013441442501

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   9
2   |   1   45    1701
3   |   1   405   134865    97135605
4   |   1   3321  10766601  70618411521    463255079498001
...
		

Links

• María Merino, Rows n=0..33 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, A286919.

Formula

For even n and m: T(n,m) = (9^(m*n) + 3*9^(m*n/2))/4;

for even n and odd m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 2*9^(m*n/2))/4;

for odd n and even m: T(n,m) = (9^(m*n) + 9^((m*n+m)/2) + 2*9^(m*n/2))/4;

for odd n and m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 9^((m*n+m)/2) + 9^((m*n+1)/2))/4.