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A356737
Number of integer partitions of n into odd parts covering an interval of odd numbers.
1
1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 8, 9, 10, 13, 13, 15, 17, 19, 21, 25, 26, 29, 33, 37, 40, 46, 49, 54, 61, 66, 72, 81, 87, 97, 106, 115, 125, 139, 150, 163, 179, 193, 210, 232, 248, 269, 293, 317, 343, 373, 401, 433, 470, 507, 545, 590, 633, 682, 737, 790
OFFSET
0,4
EXAMPLE
The a(1) = 1 through a(9) = 6 partitions:
1 11 3 31 5 33 7 53 9
111 1111 311 3111 331 3311 333
11111 111111 31111 311111 531
1111111 11111111 33111
3111111
111111111
MATHEMATICA
nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&nogapQ[(#+1)/2]&]], {n, 0, 30}]
CROSSREFS
The strict case is A034178, for compositions A332032.
The initial case is A053251, ranked by A356232 and A356603.
The initial case for compositions is A356604.
The version for compositions is A356605, ranked by A060142 /\ A356841.
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists gapless numbers, initial A055932.
Sequence in context: A317908 A056039 A181972 * A058747 A034322 A362612
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 03 2022
STATUS
approved