A286919 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 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, 8, 1, 36, 1072, 1, 288, 66816, 33693696, 1, 2080, 4197376, 17184194560, 70368756760576, 1, 16640, 268517376, 8796399206400, 288230393868451840, 9444732983468915425280, 1, 131328, 17180065792, 4503616874348544, 1180591620768950910976, 309485009825866260538195968, 81129638414606695206587887255552

Offset

0,3

Comments

Computed using Burnsides orbit-counting lemma.

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).

Formula

For even n and m: T(n,m) = (8^(m*n) + 3*8^(m*n/2))/4;

for even n and odd m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 2*8^(m*n/2))/4;

for odd n and even m: T(n,m) = (8^(m*n) + 8^((m*n+m)/2) + 2*8^(m*n/2))/4;

for odd n and m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 8^((m*n+m)/2) + 8^((m*n+1)/2))/4.

Example

Triangle begins:
========================================================
n\m |   0   1      2        3             4
----|---------------------------------------------------
0   |   1
1   |   1   8
2   |   1   36     1072
3   |   1   288    66816    33693696
4   |   1   2080   4197376  17184194560   70368756760576
...

Crossrefs

Cf. A225910, A283432, A283433, A283434, A286893, A286895.

Sequence in context: A351242 A050302 A341739 * A286260 A226374 A050401

Adjacent sequences: A286916 A286917 A286918 * A286920 A286921 A286922

Keyword

nonn,tabl

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Status

approved