A283433 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 4 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, 4, 1, 10, 76, 1, 40, 1120, 67840, 1, 136, 16576, 4212736, 1073790976, 1, 544, 263680, 268779520, 274882625536, 281475530358784, 1, 2080, 4197376, 17184194560, 70368756760576, 288230393868451840, 1180591620768950910976

Offset

0,3

Comments

Computed using Burnside's orbit-counting lemma.

Links

María Merino, Rows n=0..41 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) = (4^(m*n) + 3*4^(m*n/2))/4;

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

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

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

Example

Triangle begins:
=======================================================================
n\m |  0   1       2        3           4               5
----|------------------------------------------------------------------
0   |  1
1   |  1    4
2   |  1    10     76
3   |  1    40     1120     67840
4   |  1    136    16576    4212736     1073790976
5   |  1    544    263680   268779520   274882625536    281475530358784
...

Crossrefs

Cf. A225910, A283432.

Sequence in context: A019128 A296419 A301390 * A121463 A244608 A115022

Adjacent sequences: A283430 A283431 A283432 * A283434 A283435 A283436

Keyword

nonn,tabl

Author

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

Status

approved