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 A353403 #7 May 16 2022 17:24:20
%S A353403 1,1,0,0,3,2,5,12,16,30,45,94,159,285,477,864,1487,2643
%N A353403 Number of compositions of n whose own reversed run-lengths are a subsequence (not necessarily consecutive).
%e A353403 The a(0) = 1 through a(7) = 12 compositions:
%e A353403 () (1) . . (22) (1121) (1113) (1123)
%e A353403 (112) (1211) (1122) (1132)
%e A353403 (211) (1221) (2311)
%e A353403 (2211) (3211)
%e A353403 (3111) (11131)
%e A353403 (11212)
%e A353403 (11221)
%e A353403 (12112)
%e A353403 (12211)
%e A353403 (13111)
%e A353403 (21121)
%e A353403 (21211)
%t A353403 Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],MemberQ[Subsets[#],Reverse[Length/@Split[#]]]&]],{n,0,15}]
%Y A353403 The non-reversed version is A353390, ranked by A353402, partitions A325702.
%Y A353403 The non-reversed recursive version is A353391, ranked by A353431.
%Y A353403 The non-reversed consecutive case is A353392, ranked by A353432.
%Y A353403 The non-reversed recursive consecutive version is A353430.
%Y A353403 A003242 counts anti-run compositions, ranked by A333489.
%Y A353403 A011782 counts compositions.
%Y A353403 A169942 counts Golomb rulers, ranked by A333222.
%Y A353403 A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
%Y A353403 A325705 counts partitions containing all of their distinct multiplicities.
%Y A353403 A329739 counts compositions with all distinct run-lengths, for runs A351013.
%Y A353403 Cf. A005811, A032020, A103295, A114640, A165413, A324572, A333755, A353400, A353401, A353426.
%K A353403 nonn,more
%O A353403 0,5
%A A353403 _Gus Wiseman_, May 15 2022