This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325689 #14 Feb 15 2022 10:50:07
%S A325689 0,0,0,1,0,6,4,15,12,28,24,45,40,66,60,91,84,120,112,153,144,190,180,
%T A325689 231,220,276,264,325,312,378,364,435,420,496,480,561,544,630,612,703,
%U A325689 684,780,760,861,840,946,924,1035,1012,1128,1104,1225,1200,1326,1300,1431
%N A325689 Number of length-3 compositions of n such that no part is the sum of the other two.
%C A325689 A composition of n is a finite sequence of positive integers summing to n.
%C A325689 Confirmed recurrence relation from _Colin Barker_ for n <= 5000. - _Fausto A. C. Cariboni_, Feb 15 2022
%H A325689 Fausto A. C. Cariboni, <a href="/A325689/b325689.txt">Table of n, a(n) for n = 0..5000</a>
%F A325689 Conjectures from _Colin Barker_, May 16 2019: (Start)
%F A325689 G.f.: x^3*(1 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.
%F A325689 a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.
%F A325689 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A325689 (End)
%e A325689 The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):
%e A325689 (111) (113) (114) (115) (116)
%e A325689 (122) (141) (124) (125)
%e A325689 (131) (222) (133) (152)
%e A325689 (212) (411) (142) (161)
%e A325689 (221) (151) (215)
%e A325689 (311) (214) (233)
%e A325689 (223) (251)
%e A325689 (232) (323)
%e A325689 (241) (332)
%e A325689 (313) (512)
%e A325689 (322) (521)
%e A325689 (331) (611)
%e A325689 (412)
%e A325689 (421)
%e A325689 (511)
%t A325689 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]
%Y A325689 Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.
%Y A325689 Cf. A325676, A325688, A325690, A325691, A325694.
%K A325689 nonn
%O A325689 0,6
%A A325689 _Gus Wiseman_, May 15 2019