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

Author

Gus Wiseman, Aug 31 2022

Keywords

• nonn

Examples

			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)
		

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.

Programs

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}]

Extensions

More terms from Alois P. Heinz, Sep 01 2022