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A286919
Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 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.
3
1, 1, 8, 1, 36, 1072, 1, 288, 66816, 33693696, 1, 2080, 4197376, 17184194560, 70368756760576, 1, 16640, 268517376, 8796399206400, 288230393868451840, 9444732983468915425280, 1, 131328, 17180065792, 4503616874348544, 1180591620768950910976, 309485009825866260538195968, 81129638414606695206587887255552
OFFSET
0,3
COMMENTS
Computed using Burnsides orbit-counting lemma.
LINKS
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
FORMULA
For even n and m: T(n,m) = (8^(m*n) + 3*8^(m*n/2))/4;
for even n and odd m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 2*8^(m*n/2))/4;
for odd n and even m: T(n,m) = (8^(m*n) + 8^((m*n+m)/2) + 2*8^(m*n/2))/4;
for odd n and m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 8^((m*n+m)/2) + 8^((m*n+1)/2))/4.
EXAMPLE
Triangle begins:
========================================================
n\m | 0 1 2 3 4
----|---------------------------------------------------
0 | 1
1 | 1 8
2 | 1 36 1072
3 | 1 288 66816 33693696
4 | 1 2080 4197376 17184194560 70368756760576
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017
STATUS
approved