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%I M0295 N0106 #55 Jul 06 2023 20:41:39
%S 0,0,0,1,1,2,2,3,7,15,12,30,8,32,164,21,114,867,3226,720,4414,24412,
%T 4079,31454,3040,25737,252727,20505,191778,2140186,14554796,1669221,
%U 17754992,148553131,14708525,177117401,10567748,138584026,1953134982,103372655,1431596750,22374792451,218018425976,16852166906,254094892254
%N Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n} with X + Y = Z (as in A002849), with the property that n is in one of the subsets.
%D R. K. Guy, "Sedlacek's Conjecture on Disjoint Solutions of x+y= z," in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
%D 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.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: <a href="/A002572/a002572.jpg">front</a>, <a href="/A002572/a002572_1.jpg">back</a> [Annotated scanned copy, with permission]
%H R. K. Guy, <a href="/A002848/a002848.pdf">Sedlacek's Conjecture on Disjoint Solutions of x+y= z</a>, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
%H Christian Hercher and Frank Niedermeyer, <a href="https://arxiv.org/abs/2307.00303">Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z</a>, arXiv:2307.00303 [math.CO], 2023.
%H Nigel Martin, <a href="http://dx.doi.org/10.1016/0012-365X(94)90168-6">Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs</a>, Discrete Mathematics, Volume 125, 1994, pp. 273-277.
%F For n >= 2, a(n) = A002849(n) if n == 0,3,7,10 (mod 12), otherwise a(n) = A002849(n) - A002849(n-1). - __Franklin T. Adams-Watters_; corrected by _Max Alekseyev_, Jul 06 2023
%e Examples from _Alois P. Heinz_, Feb 12 2010:
%e A002848(7) = 3:
%e [1, 3, 4], [2, 5, 7]
%e [1, 5, 6], [3, 4, 7]
%e [2, 3, 5], [1, 6, 7]
%e A002848(8) = 7:
%e [1, 3, 4], [2, 6, 8]
%e [1, 4, 5], [2, 6, 8]
%e [1, 6, 7], [3, 5, 8]
%e [2, 3, 5], [1, 7, 8]
%e [2, 4, 6], [1, 7, 8]
%e [2, 4, 6], [3, 5, 8]
%e [3, 4, 7], [2, 6, 8]
%e A002848(10) = 12:
%e [1, 4, 5], [2, 6, 8], [3, 7, 10]
%e [1, 4, 5], [3, 6, 9], [2, 8, 10]
%e [1, 5, 6], [3, 4, 7], [2, 8, 10]
%e [1, 6, 7], [4, 5, 9], [2, 8, 10]
%e [1, 7, 8], [2, 3, 5], [4, 6, 10]
%e [1, 8, 9], [2, 3, 5], [4, 6, 10]
%e [1, 8, 9], [2, 4, 6], [3, 7, 10]
%e [1, 8, 9], [2, 5, 7], [4, 6, 10]
%e [2, 4, 6], [3, 5, 8], [1, 9, 10]
%e [2, 6, 8], [3, 4, 7], [1, 9, 10]
%e [2, 6, 8], [4, 5, 9], [3, 7, 10]
%e [2, 7, 9], [3, 5, 8], [4, 6, 10]
%e See A002849 for further examples.
%Y Cf. A002849, A108235, A161826.
%K nonn
%O 0,6
%A _N. J. A. Sloane_
%E 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
%E a(32)-a(39) from _Max Alekseyev_, Feb 23 2012
%E Definition corrected by _Max Alekseyev_, Nov 16 2012, Jul 06 2023
%E a(40)-a(42) from _Fausto A. C. Cariboni_, Mar 12 2017
%E a(43)-a(44) computed from A002849 by _Max Alekseyev_, Jul 06 2023
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