A202705 Number of irreducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms.

1, 1, 1, 2, 6, 25, 115, 649, 4046, 29674, 228030, 1987700, 18402704, 188255116, 2030067605, 23829298479, 293949166112, 3909410101509, 54360507919179, 806312701922676

Offset: 0

Author

N. J. A. Sloane, Dec 26 2011

Keywords

• nonn
• more

Comments

"Irreducible" means that there is no j such that the first j of the triples are a partition of 1, ..., 3j.

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 "K".
• R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).

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

G.f.: 2 - 1/g where g is g.f. for A104429. [corrected by Martin Fuller, Jul 08 2025]

a(n) = A279197(n) + 2*A279198(n) for n>0.

Extensions

a(11)-a(14) from Alois P. Heinz, Dec 28 2011

a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017

a(18)-a(19) from Martin Fuller, Jul 08 2025