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A286893 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 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 A286893 #25 Apr 29 2019 05:41:59
%S A286893 1,1,6,1,21,351,1,126,12096,2544696,1,666,420876,544638816,
%T A286893 705278736576,1,3996,15132096,117564302016,914040184444416,
%U A286893 7107572245840091136,1,23436,544230576,25390538401536,1184595336212990976,55268479955808421134336,2578606199622710056510488576
%N A286893 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 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 A286893 Computed using Burnside's orbit-counting lemma.
%H A286893 María Merino, <a href="/A286893/b286893.txt">Rows n=0..36 of triangle, flattened</a>
%H A286893 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 A286893 For even n and m: T(n,m) = (6^(m*n) + 3*6^(m*n/2))/4;
%F A286893 for even n and odd m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 2*6^(m*n/2))/4;
%F A286893 for odd n and even m: T(n,m) = (6^(m*n) + 6^((m*n+m)/2) + 2*6^(m*n/2))/4;
%F A286893 for odd n and m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 6^((m*n+m)/2) + 6^((m*n+1)/2))/4.
%e A286893 Triangle begins:
%e A286893 ============================================================================
%e A286893 n\m |  0  1     2         3             4                5
%e A286893 ----|-----------------------------------------------------------------------
%e A286893 0   |  1
%e A286893 1   |  1  6
%e A286893 2   |  1  21    351
%e A286893 3   |  1  126   12096     2544696
%e A286893 4   |  1  666   420876    544638816     705278736576
%e A286893 5   |  1  3996  15132096  117564302016  914040184444416  7107572245840091136
%e A286893 ...
%Y A286893 Cf. A225910, A283432, A283433, A283434.
%K A286893 nonn,tabl
%O A286893 0,3
%A A286893 _María Merino_, Imanol Unanue, _Yosu Yurramendi_, May 15 2017