cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A356605 Number of integer compositions of n into odd parts covering an interval of odd positive integers.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Aug 31 2022

Keywords

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

A332031 G.f.: Sum_{k>=1} k! * x^(k^2) / (1 - x^k).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 7, 3, 1, 9, 1, 3, 7, 27, 1, 9, 1, 27, 7, 3, 1, 33, 121, 3, 7, 27, 1, 129, 1, 27, 7, 3, 121, 753, 1, 3, 7, 147, 1, 729, 1, 27, 127, 3, 1, 753, 5041, 123, 7, 27, 1, 729, 121, 5067, 7, 3, 1, 873, 1, 3, 5047, 40347, 121, 729, 1, 27, 7, 5163, 1, 41073, 1, 3, 127
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 05 2020

Keywords

Comments

Number of compositions (ordered partitions) of n into distinct parts where either all parts are odd or all parts are even, and where every odd part or even part between the largest and smallest appears.
Number of compositions of n that are either singular compositions (just [n]), or where the difference between successive parts is always 2. - Antti Karttunen, Dec 15 2021

Examples

			a(12) = 9 because we have [12], [7, 5], [6, 4, 2], [6, 2, 4], [5, 7], [4, 6, 2], [4, 2, 6], [2, 6, 4] and [2, 4, 6].

Links

Crossrefs

Cf. A000142, A008578 (positions of 1's), A038548, A066839, A107461.
Coincides with A332032 on odd numbers.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Sum[k! x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A332031(n) = sumdiv(n, d, (d<=(n/d)) * d!); \\ Antti Karttunen, Dec 15 2021

Formula

From Antti Karttunen, Dec 15 2021: (Start)
a(n) = Sum_{d|n, d <= n/d} d!.
a(2n-1) = A332032(2n-1) for all n >= 1.
(End)

A356737 Number of integer partitions of n into odd parts covering an interval of odd numbers.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Sep 03 2022

Keywords

Examples

			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

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.
Cf. A001227, A066205, A107428, A107429, A333217, A356224, A356846.

Programs

  • 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}]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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