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

0,4

Comments

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

Links

Table of n, a(n) for n=0..52.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

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

Example

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)

Mathematica

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

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.

Sequence in context: A271974 A050824 A323703 * A323434 A323431 A211516

Adjacent sequences: A333144 A333145 A333146 * A333148 A333149 A333150

Keyword

nonn

Author

Gus Wiseman, May 16 2020

Status

approved