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 A354909 #12 Jun 22 2022 09:26:17
%S A354909 0,0,1,1,3,7,16,33,74,155,329,688,1439,2975,6154,12654,25964,53091,
%T A354909 108369,220643,448520
%N A354909 Number of integer compositions of n that are not the run-sums of any other composition.
%C A354909 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 A354909 The a(0) = 0 through a(6) = 16 compositions:
%e A354909 . . (11) (111) (112) (113) (114)
%e A354909 (211) (311) (411)
%e A354909 (1111) (1112) (1113)
%e A354909 (1121) (1122)
%e A354909 (1211) (1131)
%e A354909 (2111) (1221)
%e A354909 (11111) (1311)
%e A354909 (2112)
%e A354909 (2211)
%e A354909 (3111)
%e A354909 (11112)
%e A354909 (11121)
%e A354909 (11211)
%e A354909 (12111)
%e A354909 (21111)
%e A354909 (111111)
%t A354909 Table[Length[Complement[Join@@Permutations/@IntegerPartitions[n], Total/@Split[#]&/@Join@@Permutations/@IntegerPartitions[n]]],{n,0,15}]
%Y A354909 The version for binary words is A000918, complement A000126.
%Y A354909 These compositions are ranked by A354904 = positions of zeros in A354578.
%Y A354909 The complement is counted by A354910, ranked by A354912.
%Y A354909 A003242 counts anti-run compositions, ranked by A333489.
%Y A354909 A238279 and A333755 count compositions by number of runs.
%Y A354909 A353851 counts compositions with all equal run-sums, ranked by A353848.
%Y A354909 A353853-A353859 pertain to composition run-sum trajectory.
%Y A354909 A353932 lists run-sums of standard compositions, rows ranked by A353847.
%Y A354909 Cf. A005811, A027336, A066099, A124767, A274174, A351597, A353849, A353850, A353860, A354905, A354907.
%K A354909 nonn,more
%O A354909 0,5
%A A354909 _Gus Wiseman_, Jun 19 2022