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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.

%I A283432 #40 May 20 2023 13:39:37
%S A283432 1,1,3,1,6,27,1,18,216,5346,1,45,1701,134865,10766601,1,135,15066,
%T A283432 3608550,871858485,211829725395,1,378,133407,96997824,70607782701,
%U A283432 51472887053238,37523659114815147,1,1134,1198476,2616461190,5719211266905,12507889858389450,27354747358715650524,59824832319304600777362
%N 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.
%C A283432 Computed using Burnside's orbit-counting lemma.
%H A283432 María Merino, <a href="/A283432/b283432.txt">Rows n=0..46 of triangle, flattened</a>
%H A283432 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 A283432 For even n and m: T(n,m) = (3^(m*n) + 3*3^(m*n/2))/4;
%F A283432 for even n and odd m: T(n,m) = (3^(m*n) + 3^((m*n+n)/2) + 2*3^(m*n/2))/4;
%F A283432 for odd n and even m: T(n,m) = (3^(m*n) + 3^((m*n+m)/2) + 2*3^(m*n/2))/4;
%F A283432 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 A283432 Triangle begins:
%e A283432 ===========================================================
%e A283432 n\ m |   0  1     2      3         4           5
%e A283432 -----|-----------------------------------------------------
%e A283432 0    |   1
%e A283432 1    |   1  3
%e A283432 2    |   1  6     27
%e A283432 3    |   1  18    216    5346
%e A283432 4    |   1  45    1701   134865    10766601
%e A283432 5    |   1  135   15066  3608550   871858485   211829725395
%e A283432 ...
%t A283432 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 A283432 Cf. A225910.
%K A283432 nonn,tabl
%O A283432 0,3
%A A283432 _María Merino_, Imanol Unanue, _Yosu Yurramendi_, May 15 2017