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A283434 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 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 A283434 #30 Apr 29 2019 05:21:41
%S A283434 1,1,5,1,15,175,1,75,4125,496875,1,325,98125,61140625,38147265625,1,
%T A283434 1625,2446875,7632421875,23841923828125,74505821533203125,1,7875,
%U A283434 61046875,953736328125,14901161376953125,232830644622802734375,3637978807094573974609375
%N A283434 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 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 A283434 Computed using Burnside's orbit-counting lemma.
%H A283434 María Merino, <a href="/A283434/b283434.txt">Rows n=0..38 of triangle, flattened</a>
%H A283434 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 A283434 For even n and m: T(n,m) = (5^(m*n) + 3*5^(m*n/2))/4;
%F A283434 for even n and odd m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 2*5^(m*n/2))/4;
%F A283434 for odd n and even m: T(n,m) = (5^(m*n) + 5^((m*n+m)/2) + 2*5^(m*n/2))/4;
%F A283434 for odd n and m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 5^((m*n+m)/2) + 5^((m*n+1)/2))/4.
%e A283434 Triangle begins:
%e A283434 ============================================================================
%e A283434 n\m |   0   1      2         3            4                5
%e A283434 ----|-----------------------------------------------------------------------
%e A283434 0   |   1
%e A283434 1   |   1   5
%e A283434 2   |   1   15     175
%e A283434 3   |   1   75     4125      496875
%e A283434 4   |   1   325    98125     61140625     38147265625
%e A283434 5   |   1   1625   2446875   7632421875   23841923828125   74505821533203125
%e A283434 ...
%Y A283434 Cf. A225910, A283432, A283433.
%K A283434 nonn,tabl
%O A283434 0,3
%A A283434 _María Merino_, Imanol Unanue, _Yosu Yurramendi_, May 15 2017