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Search: a282618 -id:a282618
Displaying 1-4 of 4 results found.
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A282615
Number of self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
+10
10
0, 1, 1, 3, 4, 9, 20, 35, 102, 160, 736, 930, 5972, 6766, 59017, 61814, 671651
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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 |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |
LINKS
Table of n, a(n) for n=1..17.
FORMULA
a(n) = A282616(n) - A279197(n).
a(n) = A279199(n) - A282618(n).
EXAMPLE
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).
CROSSREFS
Cf. A104429, A202705, A279197, A279199, A282616, A282617, A282618, A282619.
All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
KEYWORD
nonn,more
AUTHOR
Peter Kagey, Feb 19 2017
EXTENSIONS
a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
STATUS
approved
A282616
Number of self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
+10
5
1, 2, 3, 5, 15, 20, 75, 93, 588, 602, 4954, 4854, 51068, 48779, 597554, 567644, 8039742
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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 |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |
LINKS
Table of n, a(n) for n=1..17.
FORMULA
a(n) = A282615(n) + A279197(n).
a(n) = A104429(n) - A282619(n).
EXAMPLE
For n = 3 the a(3) = 3 solutions are:
(7,9,8),(4,6,5),(1,3,2),
(3,9,6),(2,8,5),(1,7,4), and
(6,8,7),(2,4,3),(1,9,5).
CROSSREFS
Cf. A104429, A202705, A279197, A279199, A282615, A282617, A282618, A282619.
KEYWORD
nonn,more
AUTHOR
Peter Kagey, Feb 19 2017
EXTENSIONS
a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
STATUS
approved
A282617
Number of non-self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
+10
5
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)
OFFSET
1,4
COMMENTS
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 |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |
LINKS
Table of n, a(n) for n=1..17.
FORMULA
a(n) = A282619(n) - A282618(n).
a(n) = A202705(n) - A279197(n).
EXAMPLE
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).
CROSSREFS
Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282618, A282619.
KEYWORD
nonn,more
AUTHOR
Peter Kagey, Feb 19 2017
EXTENSIONS
a(10)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
STATUS
approved
A282619
Number of non-self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
+10
5
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)
OFFSET
1,3
COMMENTS
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 |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |
LINKS
Table of n, a(n) for n=1..17.
FORMULA
a(n) = A282617(n) + A282618(n).
a(n) = A104429(n) - A282616(n).
EXAMPLE
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).
CROSSREFS
Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282617, A282618.
KEYWORD
nonn,more
AUTHOR
Peter Kagey, Feb 19 2017
EXTENSIONS
a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
STATUS
approved
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