A283432 - OEIS

%I #40 May 20 2023 13:39:37

%S 1,1,3,1,6,27,1,18,216,5346,1,45,1701,134865,10766601,1,135,15066,

%T 3608550,871858485,211829725395,1,378,133407,96997824,70607782701,

%U 51472887053238,37523659114815147,1,1134,1198476,2616461190,5719211266905,12507889858389450,27354747358715650524,59824832319304600777362

%N 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.

%C Computed using Burnside's orbit-counting lemma.

%H María Merino, <a href="/A283432/b283432.txt">Rows n=0..46 of triangle, flattened</a>

%H M. Merino and I. Unanue, <a href="https://doi.org/10.1387/ekaia.17851">Counting squared grid patterns with Pólya Theory</a>, EKAIA, 34 (2018), 289-316 (in Basque).

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

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

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

%F 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.

%e Triangle begins:

%e ===========================================================

%e n\ m |   0  1     2      3         4           5

%e -----|-----------------------------------------------------

%e 0    |   1

%e 1    |   1  3

%e 2    |   1  6     27

%e 3    |   1  18    216    5346

%e 4    |   1  45    1701   134865    10766601

%e 5    |   1  135   15066  3608550   871858485   211829725395

%e ...

%t 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 *)

%Y Cf. A225910.

%K nonn,tabl

%O 0,3

%A _María Merino_, Imanol Unanue, _Yosu Yurramendi_, May 15 2017