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

0,3

Comments

Computed using Burnside's orbit-counting lemma.

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).

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.

Example

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
...

Crossrefs

Cf. A225910, A283432, A283433.

Sequence in context: A157385 A157397 A157405 * A019429 A221364 A211805

Adjacent sequences: A283431 A283432 A283433 * A283435 A283436 A283437

Keyword

nonn,tabl

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Status

approved