A356605 Number of integer compositions of n into odd parts covering an interval of odd positive integers.

1, 1, 1, 2, 3, 5, 6, 10, 15, 26, 41, 65, 104, 164, 262, 424, 687, 1112, 1792, 2898, 4677, 7556, 12197, 19699, 31836, 51466, 83234, 134593, 217674, 352057, 569452, 921165, 1490173, 2410784, 3900288, 6310436, 10210358, 16521108, 26733020, 43258086, 69999295

Offset

0,4

Links

Table of n, a(n) for n=0..40.

Example

The a(1) = 1 through a(8) = 15 compositions:
  (1)  (11)  (3)    (13)    (5)      (33)      (7)        (35)
             (111)  (31)    (113)    (1113)    (133)      (53)
                    (1111)  (131)    (1131)    (313)      (1133)
                            (311)    (1311)    (331)      (1313)
                            (11111)  (3111)    (11113)    (1331)
                                     (111111)  (11131)    (3113)
                                               (11311)    (3131)
                                               (13111)    (3311)
                                               (31111)    (111113)
                                               (1111111)  (111131)
                                                          (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)

Mathematica

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m], Max[m]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@OddQ/@#&&nogapQ[(#+1)/2]&]], {n, 0, 15}]

Crossrefs

These compositions are ranked by the intersection of A060142 and A356841.

Before restricting to odds we have A107428, initial A107429.

The not necessarily gapless version is A324969 (essentially A000045).

The strict case is A332032.

The initial case is A356604.

The case of partitions is A356737, initial A053251 (ranked by A356232).

A000041 counts partitions, compositions A011782.

A066208 lists numbers with all odd prime indices, counted by A000009.

A073491 lists numbers with gapless prime indices, initial A055932.

Cf. A001227, A066205, A137921, A333217, A356224, A356846.

Sequence in context: A018594 A018626 A018305 * A018693 A018255 A194358

Adjacent sequences: A356602 A356603 A356604 * A356606 A356607 A356608

Keyword

nonn

Author

Gus Wiseman, Aug 31 2022

Extensions

More terms from Alois P. Heinz, Sep 01 2022

Status

approved