A356604 - OEIS

A356604

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

%I #10 Sep 01 2022 19:48:14

%S 1,1,1,1,3,4,5,9,13,24,40,61,101,160,257,415,679,1103,1774,2884,4656,

%T 7517,12165,19653,31753,51390,83134,134412,217505,351814,569081,

%U 920769,1489587,2409992,3899347,6309059,10208628,16518910,26729830,43254212,69994082

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

%e The a(1) = 1 through a(8) = 13 compositions:

%e (1) (11) (111) (13) (113) (1113) (133) (1133)

%e (31) (131) (1131) (313) (1313)

%e (1111) (311) (1311) (331) (1331)

%e (11111) (3111) (11113) (3113)

%e (111111) (11131) (3131)

%e (11311) (3311)

%e (13111) (111113)

%e (31111) (111131)

%e (1111111) (111311)

%e (113111)

%e (131111)

%e (311111)

%e (11111111)

%e The a(9) = 24 compositions:

%e (135) (11133) (1111113) (111111111)

%e (153) (11313) (1111131)

%e (315) (11331) (1111311)

%e (351) (13113) (1113111)

%e (513) (13131) (1131111)

%e (531) (13311) (1311111)

%e (31113) (3111111)

%e (31131)

%e (31311)

%e (33111)

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[(#+1)/2]&]],{n,0,15}]

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

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

%Y This is the odd initial case of A107428.

%Y This is the odd restriction of A107429.

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

%Y The non-initial version is A356605.

%Y A000041 counts partitions, compositions A011782.

%Y A055932 lists numbers with prime indices covering an initial interval.

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

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Aug 30 2022

%E More terms from _Alois P. Heinz_, Sep 01 2022