A325689 - OEIS

%I #14 Feb 15 2022 10:50:07

%S 0,0,0,1,0,6,4,15,12,28,24,45,40,66,60,91,84,120,112,153,144,190,180,

%T 231,220,276,264,325,312,378,364,435,420,496,480,561,544,630,612,703,

%U 684,780,760,861,840,946,924,1035,1012,1128,1104,1225,1200,1326,1300,1431

%N Number of length-3 compositions of n such that no part is the sum of the other two.

%C A composition of n is a finite sequence of positive integers summing to n.

%C Confirmed recurrence relation from _Colin Barker_ for n <= 5000. - _Fausto A. C. Cariboni_, Feb 15 2022

%H Fausto A. C. Cariboni, <a href="/A325689/b325689.txt">Table of n, a(n) for n = 0..5000</a>

%F Conjectures from _Colin Barker_, May 16 2019: (Start)

%F G.f.: x^3*(1 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.

%F a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

%F (End)

%e The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):

%e   (111)  (113)  (114)  (115)  (116)

%e          (122)  (141)  (124)  (125)

%e          (131)  (222)  (133)  (152)

%e          (212)  (411)  (142)  (161)

%e          (221)         (151)  (215)

%e          (311)         (214)  (233)

%e                        (223)  (251)

%e                        (232)  (323)

%e                        (241)  (332)

%e                        (313)  (512)

%e                        (322)  (521)

%e                        (331)  (611)

%e                        (412)

%e                        (421)

%e                        (511)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]

%Y Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.

%Y Cf. A325676, A325688, A325690, A325691, A325694.

%K nonn

%O 0,6

%A _Gus Wiseman_, May 15 2019