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

1, 1, 2, 2, 11, 11, 55, 58, 486, 442, 4218, 3924, 45096, 42013, 538537, 505830, 7368091, 6959545, 111877294, 105723374, 1886636688, 1763443165, 34585786729, 32401780965, 687085545694, 642233156868, 14691047314846, 13788837896728, 340221989868538, 317342350394678, 8327884506579315

Offset: 1

Author

N. J. A. Sloane, Dec 15 2016

Keywords

• nonn

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 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.

(End)

Examples

			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)].
		

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

• 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.]

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.

Formula

a(n) = A282616(n) - A282615(n). - Martin Fuller, Jul 15 2025

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

a(18)-a(24) from Bert Dobbelaere, May 29 2025

a(25)-a(31) from Martin Fuller, Jul 15 2025