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A286921
Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 10 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.
1
1, 1, 10, 1, 55, 2575, 1, 550, 253000, 250525000, 1, 5050, 25007500, 250025500000, 2500000075000000, 1, 50500, 2500300000, 250002775000000, 25000000255000000000, 2500000000502500000000000, 1, 500500, 250000750000, 250000250500000000, 250000000000750000000000, 250000000000250500000000000000, 250000000000000000750000000000000000
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) = (10^(m*n) + 3*10^(m*n/2))/4;
for even n and odd m: T(n,m) = (10^(m*n) + 10^((m*n+n)/2) + 2*10^(m*n/2))/4;
for odd n and even m: T(n,m) = (10^(m*n) + 10^((m*n+m)/2) + 2*10^(m*n/2))/4;
for odd n and m: T(n,m) = (10^(m*n) + 10^((m*n+n)/2) + 10^((m*n+m)/2) + 10^((m*n+1)/2))/4.
EXAMPLE
Triangle begins:
==============================================================
n\m | 0 1 2 3 4
----|---------------------------------------------------------
0 | 1
1 | 1 10
2 | 1 55 2575
3 | 1 550 253000 250525000
4 | 1 5050 25007500 250025500000 2500000075000000
...
KEYWORD
nonn,tabl
AUTHOR
María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017
STATUS
approved