A286919 -id:A286919 - cp's OEIS frontend

A286920 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 9 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, 9, 1, 45, 1701, 1, 405, 134865, 97135605, 1, 3321, 10766601, 70618411521, 463255079498001, 1, 29889, 871858485, 51473762336565, 3039416437115008521, 179474497026544179696969, 1, 266085, 70607782701, 37523729625344145, 19941610769429949618201, 10597789568841677482963905405, 5632099886234793715531013441442501

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Computed using Burnsides orbit-counting lemma.

			Triangle begins:
==========================================================
n\m |   0   1     2         3              4
----|-----------------------------------------------------
0   |   1
1   |   1   9
2   |   1   45    1701
3   |   1   405   134865    97135605
4   |   1   3321  10766601  70618411521    463255079498001
...

María Merino, Rows n=0..33 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434, A286893, A286895, A286919.

For even n and m: T(n,m) = (9^(m*n) + 3*9^(m*n/2))/4;
for even n and odd m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 2*9^(m*n/2))/4;
for odd n and even m: T(n,m) = (9^(m*n) + 9^((m*n+m)/2) + 2*9^(m*n/2))/4;
for odd n and m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 9^((m*n+m)/2) + 9^((m*n+1)/2))/4.

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.

Original entry on oeis.org

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

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Computed using Burnsides orbit-counting lemma.

			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
...

María Merino, Rows n=0..32 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434, A286893, A286895, A286919, A286920.

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.

cp's OEIS Frontend

A286920 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 9 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula