A283434 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 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, 5, 1, 15, 175, 1, 75, 4125, 496875, 1, 325, 98125, 61140625, 38147265625, 1, 1625, 2446875, 7632421875, 23841923828125, 74505821533203125, 1, 7875, 61046875, 953736328125, 14901161376953125, 232830644622802734375, 3637978807094573974609375

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   5
2   |   1   15     175
3   |   1   75     4125      496875
4   |   1   325    98125     61140625     38147265625
5   |   1   1625   2446875   7632421875   23841923828125   74505821533203125
...
		

Links

• María Merino, Rows n=0..38 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.

Formula

For even n and m: T(n,m) = (5^(m*n) + 3*5^(m*n/2))/4;

for even n and odd m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 2*5^(m*n/2))/4;

for odd n and even m: T(n,m) = (5^(m*n) + 5^((m*n+m)/2) + 2*5^(m*n/2))/4;

for odd n and m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 5^((m*n+m)/2) + 5^((m*n+1)/2))/4.