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

0,5

Links

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

Example

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)

Mathematica

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

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.

Sequence in context: A242800 A136259 A099560 * A050161 A195609 A117125

Adjacent sequences: A356601 A356602 A356603 * A356605 A356606 A356607

Keyword

nonn

Author

Gus Wiseman, Aug 30 2022

Extensions

More terms from Alois P. Heinz, Sep 01 2022

Status

approved