A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

1, 1, 2, 5, 15, 55, 232, 1161, 6643, 44566, 327064, 2709050, 24312028, 240833770, 2546215687, 29251369570, 355838858402, 4658866773664

Offset

0,3

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

Table of n, a(n) for n=0..17.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]. See sequence "M".

Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.

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).

Example

{{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.

Crossrefs

Cf. A104430-A104443.

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

See also A002848, A002849, A334250.

Sequence in context: A051295 A334154 A009383 * A109319 A059219 A242275

Adjacent sequences: A104426 A104427 A104428 * A104430 A104431 A104432

Keyword

nonn,nice,more

Author

Jonas Wallgren, Mar 17 2005

Extensions

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

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

Status

approved