A286893 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 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, 6, 1, 21, 351, 1, 126, 12096, 2544696, 1, 666, 420876, 544638816, 705278736576, 1, 3996, 15132096, 117564302016, 914040184444416, 7107572245840091136, 1, 23436, 544230576, 25390538401536, 1184595336212990976, 55268479955808421134336, 2578606199622710056510488576

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  6
2   |  1  21    351
3   |  1  126   12096     2544696
4   |  1  666   420876    544638816     705278736576
5   |  1  3996  15132096  117564302016  914040184444416  7107572245840091136
...
		

Links

• María Merino, Rows n=0..36 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.

Formula

For even n and m: T(n,m) = (6^(m*n) + 3*6^(m*n/2))/4;

for even n and odd m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 2*6^(m*n/2))/4;

for odd n and even m: T(n,m) = (6^(m*n) + 6^((m*n+m)/2) + 2*6^(m*n/2))/4;

for odd n and m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 6^((m*n+m)/2) + 6^((m*n+1)/2))/4.