login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002849 Maximal number of disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.
(Formerly M0980 N0368)
9
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, 23909761856, 241928187832 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

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

Table of n, a(n) for n=1..42.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [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.

Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

EXAMPLE

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

PROG

(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(k<i1, k, if(k<i2-1, k+1, k+2))]), t-1)); break)))); r}

a(n)=nxyz(vector(n, k, k), n\3-(n%12==6|n%12==9))

\\ Franklin T. Adams-Watters

CROSSREFS

Cf. A002848, A108235, A161826.

Sequence in context: A258105 A057063 A108236 * A163234 A072984 A231655

Adjacent sequences:  A002846 A002847 A002848 * A002850 A002851 A002852

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

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

a(40)-a(41) from Fausto A. C. Cariboni, Feb 04 2017

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 29 10:48 EDT 2017. Contains 287246 sequences.