A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.

1, 1, 3, 6, 15, 23, 53, 94, 203, 404, 855, 1648, 3416, 6662, 13400, 26406, 53038, 105306, 212051, 422162, 849267, 1696864, 3406077, 6807024, 13642099, 27268122, 54576003, 109096436, 218250874, 436243705, 872533347, 1744312748, 3488432736, 6974783481

Offset

1,3

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..2000 (first 400 terms from Alois P. Heinz)

Vaclav Kotesovec, Graph a(n)/2^n

Formula

a(n) = 2^(n-1) - A240085(n). - Gus Wiseman, Nov 25 2019

Example

From Gus Wiseman, Nov 25 2019: (Start)
The a(1) = 1 through a(5) = 15 compositions:
  (1)  (2)  (3)    (4)      (5)
            (1,2)  (1,3)    (1,4)
            (2,1)  (3,1)    (2,3)
                   (1,1,2)  (3,2)
                   (1,2,1)  (4,1)
                   (2,1,1)  (1,1,3)
                            (1,2,2)
                            (1,3,1)
                            (2,1,2)
                            (2,2,1)
                            (3,1,1)
                            (1,1,1,2)
                            (1,1,2,1)
                            (1,2,1,1)
                            (2,1,1,1)
(End)

Maple

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 1) -b(n$2, 0, 2):
seq(a(n), n=1..50);

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Min@@Length/@Split[Sort[#]]==1&]], {n, 0, 10}] (* Gus Wiseman, Nov 25 2019 *)

Crossrefs

Column k=1 of A242451.

The complement is counted by A240085.

Cf. A003242, A098504, A114901, A261983, A329740, A329741.

Sequence in context: A087359 A253651 A180322 * A129602 A044888 A179805

Adjacent sequences: A244161 A244162 A244163 * A244165 A244166 A244167

Keyword

nonn

Author

Alois P. Heinz, Jun 21 2014

Status

approved