A286895 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 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, 7, 1, 28, 637, 1, 196, 30184, 10151428, 1, 1225, 1443001, 3461821825, 8308236966001, 1, 8575, 70656628, 1186972525900, 19948070175962425, 335267157313994232775, 1, 58996, 3460410037, 407106879976216, 47895307855522569001, 5634835073082541702198396, 662932711464914589254954278237

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  7
2   |  1  28   637
3   |  1  196  30184    10151428
4   |  1  1225 1443001  3461821825    8308236966001
5   |  1  8575 70656628 1186972525900 19948070175962425 335267157313994232775
...
		

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.

Formula

For even n and m: T(n,m) = (7^(m*n) + 3*7^(m*n/2))/4;

for even n and odd m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 2*7^(m*n/2))/4;

for odd n and even m: T(n,m) = (7^(m*n) + 7^((m*n+m)/2) + 2*7^(m*n/2))/4;

for odd n and m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 7^((m*n+m)/2) + 7^((m*n+1)/2))/4.