A002848 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n} with X + Y = Z (as in A002849), with the property that n is in one of the subsets.

0, 0, 0, 1, 1, 2, 2, 3, 7, 15, 12, 30, 8, 32, 164, 21, 114, 867, 3226, 720, 4414, 24412, 4079, 31454, 3040, 25737, 252727, 20505, 191778, 2140186, 14554796, 1669221, 17754992, 148553131, 14708525, 177117401, 10567748, 138584026, 1953134982, 103372655, 1431596750, 22374792451, 218018425976, 16852166906, 254094892254

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			Examples from _Alois P. Heinz_, Feb 12 2010:
A002848(7) = 3:
  [1, 3, 4], [2, 5, 7]
  [1, 5, 6], [3, 4, 7]
  [2, 3, 5], [1, 6, 7]
A002848(8) = 7:
  [1, 3, 4], [2, 6, 8]
  [1, 4, 5], [2, 6, 8]
  [1, 6, 7], [3, 5, 8]
  [2, 3, 5], [1, 7, 8]
  [2, 4, 6], [1, 7, 8]
  [2, 4, 6], [3, 5, 8]
  [3, 4, 7], [2, 6, 8]
A002848(10) = 12:
  [1, 4, 5], [2, 6, 8], [3, 7, 10]
  [1, 4, 5], [3, 6, 9], [2, 8, 10]
  [1, 5, 6], [3, 4, 7], [2, 8, 10]
  [1, 6, 7], [4, 5, 9], [2, 8, 10]
  [1, 7, 8], [2, 3, 5], [4, 6, 10]
  [1, 8, 9], [2, 3, 5], [4, 6, 10]
  [1, 8, 9], [2, 4, 6], [3, 7, 10]
  [1, 8, 9], [2, 5, 7], [4, 6, 10]
  [2, 4, 6], [3, 5, 8], [1, 9, 10]
  [2, 6, 8], [3, 4, 7], [1, 9, 10]
  [2, 6, 8], [4, 5, 9], [3, 7, 10]
  [2, 7, 9], [3, 5, 8], [4, 6, 10]
See A002849 for further examples.
		

References

• 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.
• Richard K. Guy, The unity of combinatorics, in Proc. 25th Iran. Math. Conf., Tehran, (1994), Math. Appl. 329 (1994) 129-159, Kluwer Acad. Publ., Dordrecht, 1995.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Martin Fuller, Table of n, a(n) for n = 0..52
• R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
• R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
• 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.
• Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.

Crossrefs

Cf. A002849, A108235, A161826.

Formula

For n >= 2, a(n) = A002849(n) if n == 0,3,7,10 (mod 12), otherwise a(n) = A002849(n) - A002849(n-1). - _Franklin T. Adams-Watters; corrected by Max Alekseyev, Jul 06 2023

Extensions

Edited by N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt, Max Alekseyev and others

a(32)-a(39) from Max Alekseyev, Feb 23 2012

Definition corrected by Max Alekseyev, Nov 16 2012, Jul 06 2023

a(40)-a(42) from Fausto A. C. Cariboni, Mar 12 2017

a(43)-a(44) computed from A002849 by Max Alekseyev, Jul 06 2023