A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 19, 23, 29, 35, 43, 53, 63, 75, 91, 107, 127, 151, 177, 207, 243, 283, 329, 383, 443, 511, 591, 679, 779, 895, 1023, 1169, 1335, 1519, 1727, 1963, 2225, 2519, 2851, 3219, 3631, 4095, 4607, 5179, 5819, 6527, 7315, 8193, 9163

Offset: 0

Author

Gus Wiseman, May 16 2020

Keywords

• nonn

Comments

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

Examples

			The a(1) = 1 through a(9) = 15 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)    (1,8)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)    (2,7)
                          (3,2)  (4,2)    (3,4)    (3,5)    (3,6)
                          (4,1)  (5,1)    (4,3)    (5,3)    (4,5)
                                 (1,2,3)  (5,2)    (6,2)    (5,4)
                                 (3,2,1)  (6,1)    (7,1)    (6,3)
                                          (1,2,4)  (1,2,5)  (7,2)
                                          (4,2,1)  (1,3,4)  (8,1)
                                                   (4,3,1)  (1,2,6)
                                                   (5,2,1)  (1,3,5)
                                                            (2,3,4)
                                                            (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
		

Links

• Eric Weisstein's World of Mathematics, Unimodal Sequence

Crossrefs

Strict partitions are A000009.

Unimodal compositions are A001523 (strict: A072706).

Strict compositions are A032020.

The non-strict version appears to be A329398.

Partitions with incr. or decr. run-lengths are A332745 (strict: A333190).

Compositions with incr. or decr. run-lengths are A332835 (strict: A333191).

The complement is counted by A333149 (non-strict: A332834).

Cf. A059204, A072705, A072707, A115981, A332285, A332578, A332746, A332831, A332833, A332874, A333150.

Programs

Mathematica

Table[2*PartitionsQ[n]-1,{n,0,30}]

Formula

a(n) = 2*A000009(n) - 1.