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A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

%I A279197 #53 Jul 15 2025 08:23:45
%S A279197 1,1,2,2,11,11,55,58,486,442,4218,3924,45096,42013,538537,505830,
%T A279197 7368091,6959545,111877294,105723374,1886636688,1763443165,
%U A279197 34585786729,32401780965,687085545694,642233156868,14691047314846,13788837896728,340221989868538,317342350394678,8327884506579315
%N A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
%C A279197 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 A279197 From _Peter Kagey_, Feb 14 2017: (Start)
%C A279197 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 A279197 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 A279197 (End)
%D A279197 R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
%D A279197 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 A279197 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 A279197 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 A279197 Peter Kagey, <a href="/A279197/a279197.hs.txt">Haskell program for A279197</a>.
%H A279197 Peter Kagey, <a href="/A279197/a279197.txt">Solutions for a(1)-a(10)</a>.
%H A279197 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.]
%F A279197 a(n) = A282616(n) - A282615(n). - _Martin Fuller_, Jul 15 2025
%e A279197 Examples of solutions X,Y,Z for n=5:
%e A279197   2,4,3
%e A279197   5,7,6
%e A279197   1,15,8
%e A279197   9,11,10
%e A279197   12,14,13
%e A279197 and in his letter Richard Guy has drawn links pairing the first and fifth solutions, and the second and fourth solutions.
%e A279197 For n = 2 the a(2) = 1 solution is
%e A279197   [(2,6,4),(1,5,3)].
%e A279197 For n = 3 the a(3) = 2 solutions are
%e A279197   [(1,7,4),(3,9,6),(2,8,5)] and
%e A279197   [(2,4,3),(6,8,7),(1,9,5)].
%Y A279197 All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
%Y A279197 See also A002848, A002849.
%K A279197 nonn
%O A279197 1,3
%A A279197 _N. J. A. Sloane_, Dec 15 2016
%E A279197 a(7) corrected and a(8)-a(13) added by _Peter Kagey_, Feb 14 2017
%E A279197 a(14)-a(16) from _Fausto A. C. Cariboni_, Feb 27 2017
%E A279197 a(17) from _Fausto A. C. Cariboni_, Mar 22 2017
%E A279197 a(18)-a(24) from _Bert Dobbelaere_, May 29 2025
%E A279197 a(25)-a(31) from _Martin Fuller_, Jul 15 2025