

A279197


Number of selfconjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).


10



1, 1, 2, 2, 11, 11, 55, 58, 486, 442, 4218, 3924, 45096, 42013, 538537, 505830, 7368091
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OFFSET

1,3


COMMENTS

In Richard Guy's letter, the term 50 is marked with a question mark. Peter Kagey has shown that the value should be 55.  N. J. A. Sloane, Feb 15 2017
From Peter Kagey, Feb 14 2017: (Start)
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 selfconjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (ma, mb, mc) or (mb, ma, mc) where m = 3n+1.
(End)


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. 221223.
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. 173179, 1976.


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 "I".
Peter Kagey, Haskell program for A279197.
Peter Kagey, Solutions for a(1)a(10).
R. J. Nowakowski, Generalizations of the LangfordSkolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]


EXAMPLE

Examples of solutions X,Y,Z for n=5:
2,4,3
5,7,6
1,15,8
9,11,10
12,14,13
and in his letter Richard Guy has drawn links pairing the first and fifth solutions, and the second and fourth solutions.
For n = 2 the a(2) = 1 solution is
[(2,6,4),(1,5,3)].
For n = 3 the a(3) = 2 solutions are
[(1,7,4),(3,9,6),(2,8,5)] and
[(2,4,3),(6,8,7),(1,9,5)].


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.
Sequence in context: A265530 A309477 A126806 * A121871 A194638 A327870
Adjacent sequences: A279194 A279195 A279196 * A279198 A279199 A279200


KEYWORD

nonn,more


AUTHOR

N. J. A. Sloane, Dec 15 2016


EXTENSIONS

a(7) corrected and a(8)a(13) added by Peter Kagey, Feb 14 2017
a(14)a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017


STATUS

approved



