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A286895 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 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 A286895 #27 Mar 23 2025 18:44:04
%S A286895 1,1,7,1,28,637,1,196,30184,10151428,1,1225,1443001,3461821825,
%T A286895 8308236966001,1,8575,70656628,1186972525900,19948070175962425,
%U A286895 335267157313994232775,1,58996,3460410037,407106879976216,47895307855522569001,5634835073082541702198396,662932711464914589254954278237
%N A286895 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 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 A286895 Computed using Burnside's orbit-counting lemma.
%H A286895 María Merino, <a href="/A286895/b286895.txt">Rows n=0..35 of triangle, flattened</a>
%H A286895 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 A286895 For even n and m: T(n,m) = (7^(m*n) + 3*7^(m*n/2))/4;
%F A286895 for even n and odd m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 2*7^(m*n/2))/4;
%F A286895 for odd n and even m: T(n,m) = (7^(m*n) + 7^((m*n+m)/2) + 2*7^(m*n/2))/4;
%F A286895 for odd n and m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 7^((m*n+m)/2) + 7^((m*n+1)/2))/4.
%e A286895 Triangle begins:
%e A286895 ============================================================================
%e A286895 n\m |  0  1    2        3             4                 5
%e A286895 ----|-----------------------------------------------------------------------
%e A286895 0   |  1
%e A286895 1   |  1  7
%e A286895 2   |  1  28   637
%e A286895 3   |  1  196  30184    10151428
%e A286895 4   |  1  1225 1443001  3461821825    8308236966001
%e A286895 5   |  1  8575 70656628 1186972525900 19948070175962425 335267157313994232775
%e A286895 ...
%Y A286895 Cf. A225910,  A283432, A283433, A283434, A286893.
%K A286895 nonn,tabl
%O A286895 0,3
%A A286895 _María Merino_, Imanol Unanue, _Yosu Yurramendi_, May 15 2017