A356604 Number of integer compositions of n into odd parts covering an initial interval of odd positive integers.

1, 1, 1, 1, 3, 4, 5, 9, 13, 24, 40, 61, 101, 160, 257, 415, 679, 1103, 1774, 2884, 4656, 7517, 12165, 19653, 31753, 51390, 83134, 134412, 217505, 351814, 569081, 920769, 1489587, 2409992, 3899347, 6309059, 10208628, 16518910, 26729830, 43254212, 69994082

Offset: 0

Author

Gus Wiseman, Aug 30 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (111)  (13)    (113)    (1113)    (133)      (1133)
                    (31)    (131)    (1131)    (313)      (1313)
                    (1111)  (311)    (1311)    (331)      (1331)
                            (11111)  (3111)    (11113)    (3113)
                                     (111111)  (11131)    (3131)
                                               (11311)    (3311)
                                               (13111)    (111113)
                                               (31111)    (111131)
                                               (1111111)  (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)
The a(9) = 24 compositions:
  (135)  (11133)  (1111113)  (111111111)
  (153)  (11313)  (1111131)
  (315)  (11331)  (1111311)
  (351)  (13113)  (1113111)
  (513)  (13131)  (1131111)
  (531)  (13311)  (1311111)
         (31113)  (3111111)
         (31131)
         (31311)
         (33111)
		

Crossrefs

The case of partitions is A053251, ranked by A356232 and A356603.

These compositions are ranked by the intersection of A060142 and A333217.

This is the odd initial case of A107428.

This is the odd restriction of A107429.

This is the normal/covering case of A324969 (essentially A000045).

The non-initial version is A356605.

A000041 counts partitions, compositions A011782.

A055932 lists numbers with prime indices covering an initial interval.

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

Cf. A001221, A001222, A001227, A005408, A061395, A066205, A073493, A137921, A356224.

Programs

Mathematica

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

Extensions

More terms from Alois P. Heinz, Sep 01 2022