Search: a286919 -id:a286919
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A286920
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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.
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+10
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Computed using Burnsides 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) = (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.
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EXAMPLE
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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
...
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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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A286921
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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.
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+10
1
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Computed using Burnsides 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) = (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.
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EXAMPLE
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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
...
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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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