cp's OEIS Frontend

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

%I A333147 #11 Feb 16 2025 08:33:59
%S A333147 1,1,1,3,3,5,7,9,11,15,19,23,29,35,43,53,63,75,91,107,127,151,177,207,
%T A333147 243,283,329,383,443,511,591,679,779,895,1023,1169,1335,1519,1727,
%U A333147 1963,2225,2519,2851,3219,3631,4095,4607,5179,5819,6527,7315,8193,9163
%N A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.
%C A333147 A composition of n is a finite sequence of positive integers summing to n.
%H A333147 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A333147 a(n) = 2*A000009(n) - 1.
%e A333147 The a(1) = 1 through a(9) = 15 compositions:
%e A333147   (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
%e A333147             (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)    (1,8)
%e A333147             (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)    (2,7)
%e A333147                           (3,2)  (4,2)    (3,4)    (3,5)    (3,6)
%e A333147                           (4,1)  (5,1)    (4,3)    (5,3)    (4,5)
%e A333147                                  (1,2,3)  (5,2)    (6,2)    (5,4)
%e A333147                                  (3,2,1)  (6,1)    (7,1)    (6,3)
%e A333147                                           (1,2,4)  (1,2,5)  (7,2)
%e A333147                                           (4,2,1)  (1,3,4)  (8,1)
%e A333147                                                    (4,3,1)  (1,2,6)
%e A333147                                                    (5,2,1)  (1,3,5)
%e A333147                                                             (2,3,4)
%e A333147                                                             (4,3,2)
%e A333147                                                             (5,3,1)
%e A333147                                                             (6,2,1)
%t A333147 Table[2*PartitionsQ[n]-1,{n,0,30}]
%Y A333147 Strict partitions are A000009.
%Y A333147 Unimodal compositions are A001523 (strict: A072706).
%Y A333147 Strict compositions are A032020.
%Y A333147 The non-strict version appears to be A329398.
%Y A333147 Partitions with incr. or decr. run-lengths are A332745 (strict: A333190).
%Y A333147 Compositions with incr. or decr. run-lengths are A332835 (strict: A333191).
%Y A333147 The complement is counted by A333149 (non-strict: A332834).
%Y A333147 Cf. A059204, A072705, A072707, A115981, A332285, A332578, A332746, A332831, A332833, A332874, A333150.
%K A333147 nonn
%O A333147 0,4
%A A333147 _Gus Wiseman_, May 16 2020