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 A325688 #21 Feb 15 2022 13:56:51
%S A325688 0,0,0,1,0,4,5,12,12,25,24,40,41,60,60,85,84,112,113,144,144,181,180,
%T A325688 220,221,264,264,313,312,364,365,420,420,481,480,544,545,612,612,685,
%U A325688 684,760,761,840,840,925,924,1012,1013,1104,1104,1201,1200,1300,1301,1404
%N A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.
%C A325688 A composition of n is a finite sequence of positive integers summing to n.
%C A325688 Confirmed recurrence relation from _Colin Barker_ for n <= 5000. - _Fausto A. C. Cariboni_, Feb 13 2022
%H A325688 Fausto A. C. Cariboni, <a href="/A325688/b325688.txt">Table of n, a(n) for n = 0..5000</a>
%F A325688 Conjectures from _Colin Barker_, May 16 2019: (Start)
%F A325688 G.f.: x^3*(1 + 2*x^2 + 4*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).
%F A325688 a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.
%F A325688 (End)
%e A325688 The a(3) = 1 through a(8) = 12 compositions:
%e A325688 (111) (113) (114) (115) (116)
%e A325688 (122) (132) (124) (125)
%e A325688 (221) (222) (133) (143)
%e A325688 (311) (231) (142) (152)
%e A325688 (411) (214) (215)
%e A325688 (223) (233)
%e A325688 (241) (251)
%e A325688 (322) (332)
%e A325688 (331) (341)
%e A325688 (412) (512)
%e A325688 (421) (521)
%e A325688 (511) (611)
%t A325688 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,0,30}]
%Y A325688 Column k = 3 of A325687.
%Y A325688 Cf. A000217 (all length-3).
%Y A325688 Cf. A000079, A001399, A005044, A008642, A069905, A108917, A124278, A266223.
%Y A325688 Cf. A325686, A325689, A325690, A325691.
%K A325688 nonn
%O A325688 0,6
%A A325688 _Gus Wiseman_, May 15 2019