A283432 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 3 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, 3, 1, 6, 27, 1, 18, 216, 5346, 1, 45, 1701, 134865, 10766601, 1, 135, 15066, 3608550, 871858485, 211829725395, 1, 378, 133407, 96997824, 70607782701, 51472887053238, 37523659114815147, 1, 1134, 1198476, 2616461190, 5719211266905, 12507889858389450, 27354747358715650524, 59824832319304600777362

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  3
2    |   1  6     27
3    |   1  18    216    5346
4    |   1  45    1701   134865    10766601
5    |   1  135   15066  3608550   871858485   211829725395
...
		

Links

• María Merino, Rows n=0..46 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.

Programs

Mathematica

Table[Which[AllTrue[{n,m},EvenQ],(3^(m n)+3 3^((m n)/2))/4,EvenQ[ n]&&OddQ[m],(3^(m n)+3^((m n+n)/2)+2 3^((m n)/2))/4,OddQ[n]&&EvenQ[ m],(3^(m n)+3^((m n+m)/2)+2 3^((m n)/2))/4,True,(3^(m n)+3^((m n+n)/2)+3^((m n+m)/2)+3^((m n+1)/2))/4],{n,0,10},{m,0,n}]//Flatten (* Harvey P. Dale, Mar 29 2023 *)

Formula

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

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

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

for odd n and m: T(n,m) = (3^(m*n) + 3^((m*n+n)/2) + 3^((m*n+m)/2) + 3^((m*n+1)/2))/4.