%I #43 Apr 10 2017 13:09:27

%S 1,1,2,2,11,11,55,58,486,442,4218,3924,45096,42013,538537,505830,

%T 7368091

%N Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

%C 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

%C From _Peter Kagey_, Feb 14 2017: (Start)

%C 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.)

%C 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.

%C (End)

%D R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.

%D R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.

%D 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.

%H R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: <a href="/A002572/a002572.jpg">front</a>, <a href="/A002572/a002572_1.jpg">back</a> [Annotated scanned copy, with permission] See sequence "I".

%H Peter Kagey, <a href="/A279197/a279197.hs.txt">Haskell program for A279197</a>.

%H Peter Kagey, <a href="/A279197/a279197.txt">Solutions for a(1)-a(10)</a>.

%H R. J. Nowakowski, <a href="/A104429/a104429.pdf">Generalizations of the Langford-Skolem problem</a>, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

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

%e 2,4,3

%e 5,7,6

%e 1,15,8

%e 9,11,10

%e 12,14,13

%e and in his letter Richard Guy has drawn links pairing the first and fifth solutions, and the second and fourth solutions.

%e For n = 2 the a(2) = 1 solution is

%e [(2,6,4),(1,5,3)].

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

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

%e [(2,4,3),(6,8,7),(1,9,5)].

%Y All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

%Y See also A002848, A002849.

%K nonn,more

%O 1,3

%A _N. J. A. Sloane_, Dec 15 2016

%E a(7) corrected and a(8)-a(13) added by _Peter Kagey_, Feb 14 2017

%E a(14)-a(16) from _Fausto A. C. Cariboni_, Feb 27 2017

%E a(17) from _Fausto A. C. Cariboni_, Mar 22 2017