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%I A104429 #48 Jul 05 2023 16:56:39
%S A104429 1,1,2,5,15,55,232,1161,6643,44566,327064,2709050,24312028,240833770,
%T A104429 2546215687,29251369570,355838858402,4658866773664
%N A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.
%D A104429 R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
%D A104429 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 A104429 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.
%H A104429 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]. See sequence "M".
%H A104429 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 A104429 R. J. Nowakowski, <a href="/A104429/a104429.pdf">Generalizations of the Langford-Skolem problem</a>, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).
%e A104429 {{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.
%Y A104429 Cf. A104430-A104443.
%Y A104429 All of A279197, A279198, A202705, A279199, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
%Y A104429 See also A002848, A002849, A334250.
%K A104429 nonn,nice,more
%O A104429 0,3
%A A104429 _Jonas Wallgren_, Mar 17 2005
%E A104429 a(11)-a(14) from _Alois P. Heinz_, Dec 28 2011
%E A104429 a(15)-a(17) from _Fausto A. C. Cariboni_, Feb 22 2017

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