%I #6 May 17 2022 07:24:58

%S 1,1,1,1,2,1,3,1,1,4,5,7,9,11,15,16,22,25,37,37,45

%N Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

%e The a(n) compositions for selected n (A..E = 10..14):

%e n=4: n=6: n=9: n=10: n=12: n=14:

%e -----------------------------------------------------------

%e (4) (6) (9) (A) (C) (E)

%e (22) (1122) (333) (2233) (2244) (2255)

%e (2211) (121122) (3322) (4422) (5522)

%e (221121) (131122) (151122) (171122)

%e (221131) (221124) (221126)

%e (221142) (221135)

%e (221151) (221153)

%e (241122) (221162)

%e (421122) (221171)

%e (261122)

%e (351122)

%e (531122)

%e (621122)

%e (122121122)

%e (221121221)

%t yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yoyQ]],{n,0,15}]

%Y Non-recursive non-consecutive version: counted by A353390, ranked by A353402, reverse A353403, partitions A325702.

%Y Non-consecutive version: A353391, ranked by A353431, partitions A353426.

%Y Non-recursive version: A353392, ranked by A353432.

%Y A003242 counts anti-run compositions, ranked by A333489.

%Y A011782 counts compositions.

%Y A114901 counts compositions with no runs of length 1.

%Y A169942 counts Golomb rulers, ranked by A333222.

%Y A325676 counts knapsack compositions, ranked by A333223.

%Y A329738 counts uniform compositions, partitions A047966.

%Y A329739 counts compositions with all distinct run-lengths.

%Y Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, May 16 2022