A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

0, 0, 1, 1, 5, 11, 25, 55, 117, 241, 493, 1001, 2019, 4061, 8149, 16331, 32705, 65461, 130981, 262037, 524161, 1048425, 2096975, 4194097, 8388365, 16776933, 33554103, 67108481, 134217285, 268434945, 536870321, 1073741145, 2147482869, 4294966401, 8589933569

Offset: 0

Author

Gus Wiseman, Sep 11 2020

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n.

Examples

			The a(2) = 1 through a(5) = 11 compositions:
  (11)  (111)  (22)    (113)
               (112)   (122)
               (121)   (131)
               (211)   (212)
               (1111)  (221)
                       (311)
                       (1112)
                       (1121)
                       (1211)
                       (2111)
                       (11111)
		

Crossrefs

Ranked by the complement of the intersection of A333255 and A333256.

A332834 is the weak version.

A337482 is the semi-strict version.

A337484 counts only compositions of length 3.

A007318 and A097805 count compositions by length.

A032020 counts strict compositions, ranked by A233564.

A218004 counts strictly increasing or weakly decreasing compositions.

Cf. A216652, A329398, A337462, A337483, A337605.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!Greater@@#&]],{n,0,15}]

Formula

a(n) = 2^(n-1) - 2*A000009(n) + 1, n > 0.