Search: a282616 -id:a282616
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A282615
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Number of self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
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+10
10
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0, 1, 1, 3, 4, 9, 20, 35, 102, 160, 736, 930, 5972, 6766, 59017, 61814, 671651
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refs;
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OFFSET
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1,4
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COMMENTS
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An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
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LINKS
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FORMULA
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EXAMPLE
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For n = 4 the a(4) = 3 solutions are:
(10,12,11),(7,9,8),(4,6,5),(1,3,2),
(10,12,11),(5,9,7),(4,8,6),(1,3,2), and
(8,12,10),(7,11,9),(2,6,4),(1,5,3).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A282617
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Number of non-self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
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+10
5
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0, 0, 0, 4, 14, 104, 594, 3988, 29188, 227588, 1983482, 18398780, 188210020, 2030025592, 23828759942, 293948660282, 3909402733418
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
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LINKS
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FORMULA
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EXAMPLE
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For n = 4 the a(4) = 4 solutions are:
(7,11,9),(4,12,8),(2,10,6),(1,5,3),
(9,11,10),(4,8,6),(2,12,7),(1,5,3),
(8,12,10),(3,11,7),(2,6,4),(1,9,5), and
(8,12,10),(5,9,7),(2,4,3),(1,11,6).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A282618
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Number of non-self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
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+10
5
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0, 0, 2, 6, 26, 108, 492, 2562, 14790, 98874, 720614, 5908394, 52572682, 516141316, 5422012074, 61889630476, 749456000504
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
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LINKS
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FORMULA
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EXAMPLE
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For n = 3 the a(3) = 2 solutions are:
(5,9,7),(4,8,6),(1,3,2), and
(7,9,8),(2,6,4),(1,5,3).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A282619
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Number of non-self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
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+10
5
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0, 0, 2, 10, 40, 212, 1086, 6550, 43978, 326462, 2704096, 24307174, 240782702, 2546166908, 29250772016, 355838290758, 4658858733922
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
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LINKS
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FORMULA
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EXAMPLE
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For n = 3 the a(3) = 3 solutions are
(5,9,7),(4,8,6),(1,3,2),
(7,9,8),(2,6,4),(1,5,3).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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