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A283432
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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.
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8
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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
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OFFSET
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0,3
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COMMENTS
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Computed using Burnside's orbit-counting lemma.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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