A283434 - OEIS

%I #30 Apr 29 2019 05:21:41

%S 1,1,5,1,15,175,1,75,4125,496875,1,325,98125,61140625,38147265625,1,

%T 1625,2446875,7632421875,23841923828125,74505821533203125,1,7875,

%U 61046875,953736328125,14901161376953125,232830644622802734375,3637978807094573974609375

%N 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 Computed using Burnside's orbit-counting lemma.

%H María Merino, <a href="/A283434/b283434.txt">Rows n=0..38 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) = (5^(m*n) + 3*5^(m*n/2))/4;

%F for even n and odd m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 2*5^(m*n/2))/4;

%F for odd n and even m: T(n,m) = (5^(m*n) + 5^((m*n+m)/2) + 2*5^(m*n/2))/4;

%F 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 Triangle begins:

%e ============================================================================

%e n\m |   0   1      2         3            4                5

%e ----|-----------------------------------------------------------------------

%e 0   |   1

%e 1   |   1   5

%e 2   |   1   15     175

%e 3   |   1   75     4125      496875

%e 4   |   1   325    98125     61140625     38147265625

%e 5   |   1   1625   2446875   7632421875   23841923828125   74505821533203125

%e ...

%Y Cf. A225910, A283432, A283433.

%K nonn,tabl

%O 0,3

%A _María Merino_, Imanol Unanue, _Yosu Yurramendi_, May 15 2017