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%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
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