|
| |
|
|
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," 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, 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.
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 * A329492 A163234 A072984
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
|
| |
|
|
