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 A354910 #10 Jun 22 2022 09:27:28
%S A354910 1,1,1,3,5,9,16,31,54,101,183,336,609,1121,2038,3730,6804,12445,22703,
%T A354910 41501,75768
%N A354910 Number of compositions of n that are the run-sums of some other composition.
%C A354910 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).
%e A354910 The a(0) = 0 through a(6) = 16 compositions:
%e A354910 () (1) (2) (3) (4) (5) (6)
%e A354910 (12) (13) (14) (15)
%e A354910 (21) (22) (23) (24)
%e A354910 (31) (32) (33)
%e A354910 (121) (41) (42)
%e A354910 (122) (51)
%e A354910 (131) (123)
%e A354910 (212) (132)
%e A354910 (221) (141)
%e A354910 (213)
%e A354910 (222)
%e A354910 (231)
%e A354910 (312)
%e A354910 (321)
%e A354910 (1212)
%e A354910 (2121)
%t A354910 Table[Length[Union[Total/@Split[#]&/@ Join@@Permutations/@IntegerPartitions[n]]],{n,0,15}]
%Y A354910 The version for binary words is A000126, complement A000918
%Y A354910 The complement is counted by A354909, ranked by A354904.
%Y A354910 These compositions are ranked by A354912 = nonzeros of A354578.
%Y A354910 A003242 counts anti-run compositions, ranked by A333489.
%Y A354910 A238279 and A333755 count compositions by number of runs.
%Y A354910 A353851 counts compositions with all equal run-sums, ranked by A353848.
%Y A354910 A353853-A353859 pertain to composition run-sum trajectory.
%Y A354910 A353932 lists run-sums of standard compositions, rows ranked by A353847.
%Y A354910 Cf. A005811, A027336, A066099, A239312, A274174, A351014, A351597, A353849, A353850, A353864, A354905, A354907.
%K A354910 nonn,more
%O A354910 0,4
%A A354910 _Gus Wiseman_, Jun 20 2022