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A279197 Number of self-conjugate 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 (list; graph; refs; listen; history; text; internal format)



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 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.



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.


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 Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]


Examples of solutions X,Y,Z for n=5:






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


For n = 3 the a(3) = 2 solutions are

[(1,7,4),(3,9,6),(2,8,5)] and



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




N. J. A. Sloane, Dec 15 2016


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



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Last modified May 12 02:23 EDT 2021. Contains 343808 sequences. (Running on oeis4.)