A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

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

Offset

0,5

Links

Table of n, a(n) for n=0..20.

Example

The a(n) compositions for selected n (A..E = 10..14):
  n=4:  n=6:    n=9:      n=10:     n=12:     n=14:
-----------------------------------------------------------
  (4)   (6)     (9)       (A)       (C)       (E)
  (22)  (1122)  (333)     (2233)    (2244)    (2255)
        (2211)  (121122)  (3322)    (4422)    (5522)
                (221121)  (131122)  (151122)  (171122)
                          (221131)  (221124)  (221126)
                                    (221142)  (221135)
                                    (221151)  (221153)
                                    (241122)  (221162)
                                    (421122)  (221171)
                                              (261122)
                                              (351122)
                                              (531122)
                                              (621122)
                                              (122121122)
                                              (221121221)

Mathematica

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

Crossrefs

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

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

Non-recursive version: A353392, ranked by A353432.

A003242 counts anti-run compositions, ranked by A333489.

A011782 counts compositions.

A114901 counts compositions with no runs of length 1.

A169942 counts Golomb rulers, ranked by A333222.

A325676 counts knapsack compositions, ranked by A333223.

A329738 counts uniform compositions, partitions A047966.

A329739 counts compositions with all distinct run-lengths.

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

Sequence in context: A209344 A294099 A209115 * A353391 A141412 A178623

Adjacent sequences: A353427 A353428 A353429 * A353431 A353432 A353433

Keyword

nonn,more

Author

Gus Wiseman, May 16 2022

Status

approved