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 A329743 #6 Nov 21 2019 22:14:59
%S A329743 0,0,0,1,2,6,9,16,8
%N A329743 Number of compositions of n with runs-resistance n - 3.
%C A329743 A composition of n is a finite sequence of positive integers with sum n.
%C A329743 For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
%e A329743 The a(3) = 1 through a(8) = 8 compositions:
%e A329743 (3) (22) (14) (114) (1123) (12113)
%e A329743 (1111) (23) (411) (1132) (12212)
%e A329743 (32) (1113) (1141) (13112)
%e A329743 (41) (1221) (1411) (21131)
%e A329743 (131) (2112) (2122) (21221)
%e A329743 (212) (3111) (2212) (31121)
%e A329743 (11112) (2311) (121112)
%e A329743 (11211) (3211) (211121)
%e A329743 (21111) (11131)
%e A329743 (11212)
%e A329743 (11221)
%e A329743 (12211)
%e A329743 (13111)
%e A329743 (21211)
%e A329743 (111121)
%e A329743 (121111)
%e A329743 For example, repeatedly taking run-lengths starting with (1,2,1,1,3) gives (1,2,1,1,3) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), which is 5 steps, and 5 = 8 - 3, so (1,2,1,1,3) is counted under a(8).
%t A329743 runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
%t A329743 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-3&]],{n,10}]
%Y A329743 Column k = n - 3 of A329744.
%Y A329743 Column k = 3 of A329750.
%Y A329743 Compositions with runs-resistance 2 are A329745.
%Y A329743 Cf. A000740, A008965, A098504, A242882, A318928, A329746, A329747, A329767.
%K A329743 nonn,fini,full
%O A329743 0,5
%A A329743 _Gus Wiseman_, Nov 21 2019