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 A353392 #6 May 16 2022 17:23:51
%S A353392 1,1,0,0,1,2,2,2,2,8,12,16,20,35,46,59,81,109,144,202,282
%N A353392 Number of compositions of n whose own run-lengths are a consecutive subsequence.
%e A353392 The a(0) = 0 through a(10) = 12 compositions (empty columns indicated by dots, 0 is the empty composition):
%e A353392 0 1 . . 22 122 1122 11221 21122 333 1333
%e A353392 221 2211 12211 22112 22113 2233
%e A353392 22122 3322
%e A353392 31122 3331
%e A353392 121122 22114
%e A353392 122112 41122
%e A353392 211221 122113
%e A353392 221121 131122
%e A353392 221131
%e A353392 311221
%e A353392 1211221
%e A353392 1221121
%t A353392 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||MemberQ[Join@@Table[Take[#,{i,j}],{i,Length[#]},{j,i,Length[#]}],Length/@Split[#]]&]],{n,0,15}]
%Y A353392 The non-consecutive version for partitions is A325702.
%Y A353392 The non-consecutive version is A353390, ranked by A353402.
%Y A353392 The non-consecutive recursive version is A353391, ranked by A353431.
%Y A353392 The non-consecutive reverse version is A353403.
%Y A353392 The recursive version is A353430.
%Y A353392 These compositions are ranked by A353432.
%Y A353392 A003242 counts anti-run compositions, ranked by A333489.
%Y A353392 A011782 counts compositions.
%Y A353392 A169942 counts Golomb rulers, ranked by A333222.
%Y A353392 A325676 counts knapsack compositions, ranked by A333223.
%Y A353392 A329738 counts uniform compositions, partitions A047966.
%Y A353392 A329739 counts compositions with all distinct run-lengths.
%Y A353392 Cf. A008965, A032020, A103295, A103300, A114901, A238279, A324572, A325705, A333224, A333755, A351013, A353401.
%K A353392 nonn,more
%O A353392 0,6
%A A353392 _Gus Wiseman_, May 15 2022