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A002849
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Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.
(Formerly M0980 N0368)
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10
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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, 19424213, 167977344, 14708525, 191825926, 10567748, 149151774, 2102286756, 103372655, 1534969405
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OFFSET
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1,4
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REFERENCES
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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).
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LINKS
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R. K. Guy, Letter to N. J. A. Sloane, Jun 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.
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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