[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a279197 -id:a279197
Displaying 1-9 of 9 results found. page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms. +10
40
1, 1, 2, 5, 15, 55, 232, 1161, 6643, 44566, 327064, 2709050, 24312028, 240833770, 2546215687, 29251369570, 355838858402, 4658866773664 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
LINKS
Table of n, a(n) for n=0..17.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]. See sequence "M".
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).
EXAMPLE
{{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.
CROSSREFS
Cf. A104430-A104443.
All of A279197, A279198, A202705, A279199, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849, A334250.
KEYWORD
nonn,nice,more
AUTHOR
Jonas Wallgren, Mar 17 2005
EXTENSIONS
a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
STATUS
approved
A202705 Number of irreducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms. +10
10
1, 1, 1, 2, 6, 25, 115, 649, 4046, 29674, 228030, 1987700, 18402704, 188255116, 2030067605, 23829298479, 293949166112, 3909410101509 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
"Irreducible" means that there is no j such that the first j of the triples are a partition of 1, ..., 3j.
REFERENCES
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
LINKS
Table of n, a(n) for n=0..17.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "K".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).
FORMULA
G.f. = 1 - 1/g where g is g.f. for A104429.
a(n) = A279197(n) + 2*A279198(n) for n>0.
CROSSREFS
All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849.
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Dec 26 2011
EXTENSIONS
a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
STATUS
approved
A279199 Number of reducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms: a(n) = A104429(n) - A202705(n). +10
10
0, 0, 1, 3, 9, 30, 117, 512, 2597, 14892, 99034, 721350, 5909324, 52578654, 516148082, 5422071091, 61889692290, 749456672155 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
REFERENCES
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
LINKS
Table of n, a(n) for n=0..17.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "L".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
CROSSREFS
All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849.
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Dec 15 2016
EXTENSIONS
Definition corrected by N. J. A. Sloane, Jan 09 2017 at the suggestion of Fausto A. C. Cariboni.
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
STATUS
approved
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
A279198 Number of pairs of conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}). +10
6
0, 0, 0, 2, 7, 52, 297, 1994, 14594, 113794, 991741, 9199390, 94105010, 1015012796, 11914379971, 146974330141, 1954701366709 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
REFERENCES
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
Nowakowski, Richard Joseph, Generalization of the Langford-Skolem problem, MS Thesis, University of Calgary, 1975.
LINKS
Table of n, a(n) for n=1..17.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "J".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
FORMULA
A279197(n) + 2*A279198(n) = A202705(n).
EXAMPLE
Richard Guy gives examples in his letter.
CROSSREFS
All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849.
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Dec 15 2016
EXTENSIONS
a(7)-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
A282618 Number of non-self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}). +10
5
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)
OFFSET
1,3
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) - A282617(n).
a(n) = A279199(n) - A282615(n).
EXAMPLE
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).
CROSSREFS
Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282617, 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
page 1

Search completed in 0.010 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)