A283432 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 3 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, 3, 1, 6, 27, 1, 18, 216, 5346, 1, 45, 1701, 134865, 10766601, 1, 135, 15066, 3608550, 871858485, 211829725395, 1, 378, 133407, 96997824, 70607782701, 51472887053238, 37523659114815147, 1, 1134, 1198476, 2616461190, 5719211266905, 12507889858389450, 27354747358715650524, 59824832319304600777362

Offset

0,3

Comments

Computed using Burnside's orbit-counting lemma.

Links

María Merino, Rows n=0..46 of triangle, flattened

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) = (3^(m*n) + 3*3^(m*n/2))/4;

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

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

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

Example

Triangle begins:
===========================================================
n\ m |   0  1     2      3         4           5
-----|-----------------------------------------------------
0    |   1
1    |   1  3
2    |   1  6     27
3    |   1  18    216    5346
4    |   1  45    1701   134865    10766601
5    |   1  135   15066  3608550   871858485   211829725395
...

Mathematica

Table[Which[AllTrue[{n, m}, EvenQ], (3^(m n)+3 3^((m n)/2))/4, EvenQ[ n]&&OddQ[m], (3^(m n)+3^((m n+n)/2)+2 3^((m n)/2))/4, OddQ[n]&&EvenQ[ m], (3^(m n)+3^((m n+m)/2)+2 3^((m n)/2))/4, True, (3^(m n)+3^((m n+n)/2)+3^((m n+m)/2)+3^((m n+1)/2))/4], {n, 0, 10}, {m, 0, n}]//Flatten (* Harvey P. Dale, Mar 29 2023 *)

Crossrefs

Cf. A225910.

Sequence in context: A175291 A156363 A221929 * A157866 A221852 A363196

Adjacent sequences: A283429 A283430 A283431 * A283433 A283434 A283435

Keyword

nonn,tabl

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Status

approved