cp's OEIS Frontend

A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

%I A325686 #21 Jun 08 2025 21:43:53
%S A325686 0,0,0,0,0,0,2,6,8,18,16,30,34,48,48,72,72,96,98,126,128,162,160,198,
%T A325686 202,240,240,288,288,336,338,390,392,450,448,510,514,576,576,648,648,
%U A325686 720,722,798,800,882,880,966,970,1056,1056,1152,1152,1248,1250,1350,1352
%N A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.
%C A325686 A strict composition of n is a finite sequence of distinct positive integers summing to n.
%C A325686 From _Kevin O'Bryant_, Jun 02 2025: (Start)
%C A325686 Also the number of Sidon sets in {0,1,...,n} with 4 elements that contain both 0 and n.
%C A325686 Also, the number of 3-tuples of positive integers with the 6 numbers x, y, z, x+y, y+z, x+y+z=n all distinct. (End)
%H A325686 Fausto A. C. Cariboni, <a href="/A325686/b325686.txt">Table of n, a(n) for n = 0..5000</a>
%F A325686 Conjectures from _Colin Barker_, May 14 2019: (Start)
%F A325686 G.f.: 2*x^6*(1 + 3*x + 3*x^2 + 5*x^3) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
%F A325686 a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9. (End)
%F A325686 Above conjecture confirmed for n <= 5000. - _Fausto A. C. Cariboni_, Feb 17 2022
%e A325686 The a(6) = 2 through a(10) = 16 compositions:
%e A325686   (132)  (124)  (125)  (126)  (127)
%e A325686   (231)  (142)  (143)  (135)  (136)
%e A325686          (214)  (152)  (153)  (154)
%e A325686          (241)  (215)  (162)  (163)
%e A325686          (412)  (251)  (216)  (172)
%e A325686          (421)  (341)  (234)  (217)
%e A325686                 (512)  (243)  (253)
%e A325686                 (521)  (261)  (271)
%e A325686                        (315)  (316)
%e A325686                        (324)  (352)
%e A325686                        (342)  (361)
%e A325686                        (351)  (451)
%e A325686                        (423)  (613)
%e A325686                        (432)  (631)
%e A325686                        (513)  (712)
%e A325686                        (531)  (721)
%e A325686                        (612)
%e A325686                        (621)
%t A325686 Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z &&x!=y+z]],{n,0,30}]
%Y A325686 Column k = 3 of A325677.
%Y A325686 Cf. A000079, A001399, A005044, A108917, A124278, A143823, A266223, A275972.
%Y A325686 Cf. A325676, A325688, A325689, A325690, A325691.
%K A325686 nonn
%O A325686 0,7
%A A325686 _Gus Wiseman_, May 13 2019