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A002849
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Number of partitions of a subset of {1, 2, ..., n} into triples (X,Y,Z) each satisfying X+Y=Z, with the maximal possible number of such triples.
(Formerly M0980 N0368)
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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.
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. [From N. J. A. Sloane, Dec 30 2011]
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).
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EXAMPLE
| For m = 1 the unique solution is 1 + 2 = 3.
For m = 4 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
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PROG
| (PARI program from Franklin T. Adams-Watters)
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))
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CROSSREFS
| Cf. A002848, A108235, A161826.
Sequence in context: A122280 A057063 A108236 * A163234 A072984 A018841
Adjacent sequences: A002846 A002847 A002848 * A002850 A002851 A002852
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, Richard K. Guy, R. H. Hardin, Alois Heinz, Andrew Weimholt, Max Alekseyev and others.
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