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 A353858 #6 Jun 17 2022 22:12:40
%S A353858 0,1,2,2,5,2,8,2,20,5,8,2,78,2,8,8,223,2,179,2,142,8,8,2,4808
%N A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.
%C A353858 Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).
%e A353858 The a(0) = 0 through a(8) = 20 compositions:
%e A353858 . (1) (2) (3) (4) (5) (6) (7) (8)
%e A353858 (11) (111) (22) (11111) (33) (1111111) (44)
%e A353858 (112) (222) (224)
%e A353858 (211) (1113) (422)
%e A353858 (1111) (2112) (1124)
%e A353858 (3111) (2114)
%e A353858 (11211) (2222)
%e A353858 (111111) (4112)
%e A353858 (4211)
%e A353858 (11114)
%e A353858 (21122)
%e A353858 (22112)
%e A353858 (41111)
%e A353858 (111122)
%e A353858 (112112)
%e A353858 (211211)
%e A353858 (221111)
%e A353858 (1111211)
%e A353858 (1121111)
%e A353858 (11111111)
%t A353858 Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&,#]]==1&]],{n,0,15}]
%Y A353858 The version for partitions is A353845, ranked by A353844.
%Y A353858 The trajectory itself is A353853, last part A353855.
%Y A353858 The lengths of trajectories of standard compositions are A353854.
%Y A353858 This is column k = 1 of A353856, for partitions A353843.
%Y A353858 These compositions are ranked by A353857.
%Y A353858 A011782 counts compositions.
%Y A353858 A066099 lists compositions in standard order.
%Y A353858 A238279 and A333755 count compositions by number of runs.
%Y A353858 A275870 counts collapsible partitions, ranked by A300273.
%Y A353858 A333489 ranks anti-runs, counted by A003242 (complement A261983).
%Y A353858 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353858 A353847 represents the run-sums of a composition, partitions A353832.
%Y A353858 A353851 counts compositions with equal run-sums, ranked by A353848.
%Y A353858 A353859 counts compositions by length of run-sum trajectory.
%Y A353858 A353860 counts collapsible compositions.
%Y A353858 A353932 lists run-sums of standard compositions.
%Y A353858 Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.
%K A353858 nonn,more
%O A353858 0,3
%A A353858 _Gus Wiseman_, Jun 17 2022