A002849 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.

1, 1, 1, 2, 4, 6, 3, 10, 25, 12, 42, 8, 40, 204, 21, 135, 1002, 4228, 720, 5134, 29546, 4079, 35533, 3040, 28777, 281504, 20505, 212283, 2352469, 16907265, 1669221, 19424213, 167977344, 14708525, 191825926, 10567748, 149151774, 2102286756, 103372655, 1534969405

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			For n = 3, the unique solution is 1 + 2 = 3.
For n = 12, there are 8 solutions:
  1  5  6 | 1  5  6 | 2  5  7 | 1  6  7
  2  8 10 | 3  7 10 | 3  6  9 | 4  5  9
  4  7 11 | 2  9 11 | 1 10 11 | 3  8 11
  3  9 12 | 4  8 12 | 4  8 12 | 2 10 12
  --------+---------+---------+--------
  2  4  6 | 2  6  8 | 3  4  7 | 3  5  8
  1  9 10 | 4  5  9 | 1  8  9 | 2  7  9
  3  8 11 | 3  7 10 | 5  6 11 | 4  6 10
  5  7 12 | 1 11 12 | 2 10 12 | 1 11 12
		

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.
• 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 = 1..52 (terms 1..42 from Fausto A. C. Cariboni, terms 43..44 from Frank Niedermeyer)
• R. K. Guy, Letter to N. J. A. Sloane, Jun 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]
• 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.
• 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.
• Matheplanet Calculating sequence element a(16) of OEIS A108235
• R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Crossrefs

Cf. A002848, A108235, A161826.

Programs

PARI

nxyz(v,t)=local(n,r,x2); r=0; if(t==0,return(1)); for(i3=3*t,#v, n=v[i3]; for(i1=1,i3-2, x2=n-v[i1]; if(x2<=v[i1],break); for(i2=i1+1,i3-1, if(v[i2]>=x2, if(v[i2]==x2, r+=nxyz(vector(i3-3,k,v[if(kFranklin T. Adams-Watters

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(41) from Fausto A. C. Cariboni, Feb 04 2017

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