Search: a283433 -id:a283433
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A283434
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Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 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
6
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1, 1, 5, 1, 15, 175, 1, 75, 4125, 496875, 1, 325, 98125, 61140625, 38147265625, 1, 1625, 2446875, 7632421875, 23841923828125, 74505821533203125, 1, 7875, 61046875, 953736328125, 14901161376953125, 232830644622802734375, 3637978807094573974609375
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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) = (5^(m*n) + 3*5^(m*n/2))/4;
for even n and odd m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 2*5^(m*n/2))/4;
for odd n and even m: T(n,m) = (5^(m*n) + 5^((m*n+m)/2) + 2*5^(m*n/2))/4;
for odd n and m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 5^((m*n+m)/2) + 5^((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 5
2 | 1 15 175
3 | 1 75 4125 496875
4 | 1 325 98125 61140625 38147265625
5 | 1 1625 2446875 7632421875 23841923828125 74505821533203125
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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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A286893
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Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 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
5
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1, 1, 6, 1, 21, 351, 1, 126, 12096, 2544696, 1, 666, 420876, 544638816, 705278736576, 1, 3996, 15132096, 117564302016, 914040184444416, 7107572245840091136, 1, 23436, 544230576, 25390538401536, 1184595336212990976, 55268479955808421134336, 2578606199622710056510488576
(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 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) = (6^(m*n) + 3*6^(m*n/2))/4;
for even n and odd m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 2*6^(m*n/2))/4;
for odd n and even m: T(n,m) = (6^(m*n) + 6^((m*n+m)/2) + 2*6^(m*n/2))/4;
for odd n and m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 6^((m*n+m)/2) + 6^((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 6
2 | 1 21 351
3 | 1 126 12096 2544696
4 | 1 666 420876 544638816 705278736576
5 | 1 3996 15132096 117564302016 914040184444416 7107572245840091136
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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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A286895
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Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 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
4
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1, 1, 7, 1, 28, 637, 1, 196, 30184, 10151428, 1, 1225, 1443001, 3461821825, 8308236966001, 1, 8575, 70656628, 1186972525900, 19948070175962425, 335267157313994232775, 1, 58996, 3460410037, 407106879976216, 47895307855522569001, 5634835073082541702198396, 662932711464914589254954278237
(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 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) = (7^(m*n) + 3*7^(m*n/2))/4;
for even n and odd m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 2*7^(m*n/2))/4;
for odd n and even m: T(n,m) = (7^(m*n) + 7^((m*n+m)/2) + 2*7^(m*n/2))/4;
for odd n and m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 7^((m*n+m)/2) + 7^((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 7
2 | 1 28 637
3 | 1 196 30184 10151428
4 | 1 1225 1443001 3461821825 8308236966001
5 | 1 8575 70656628 1186972525900 19948070175962425 335267157313994232775
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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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A286919
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Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 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
3
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1, 1, 8, 1, 36, 1072, 1, 288, 66816, 33693696, 1, 2080, 4197376, 17184194560, 70368756760576, 1, 16640, 268517376, 8796399206400, 288230393868451840, 9444732983468915425280, 1, 131328, 17180065792, 4503616874348544, 1180591620768950910976, 309485009825866260538195968, 81129638414606695206587887255552
(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) = (8^(m*n) + 3*8^(m*n/2))/4;
for even n and odd m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 2*8^(m*n/2))/4;
for odd n and even m: T(n,m) = (8^(m*n) + 8^((m*n+m)/2) + 2*8^(m*n/2))/4;
for odd n and m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 8^((m*n+m)/2) + 8^((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 8
2 | 1 36 1072
3 | 1 288 66816 33693696
4 | 1 2080 4197376 17184194560 70368756760576
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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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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
2
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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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