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%I #10 May 18 2020 06:37:35
%S 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 243,283,329,383,443,511,591,679,779,895,1023,1169,1335,1519,1727,
%U 1963,2225,2519,2851,3219,3631,4095,4607,5179,5819,6527,7315,8193,9163
%N Number of compositions of n that are either strictly increasing or strictly decreasing.
%C A composition of n is a finite sequence of positive integers summing to n.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F a(n) = 2*A000009(n) - 1.
%e The a(1) = 1 through a(9) = 15 compositions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8)
%e (2,1) (3,1) (2,3) (2,4) (2,5) (2,6) (2,7)
%e (3,2) (4,2) (3,4) (3,5) (3,6)
%e (4,1) (5,1) (4,3) (5,3) (4,5)
%e (1,2,3) (5,2) (6,2) (5,4)
%e (3,2,1) (6,1) (7,1) (6,3)
%e (1,2,4) (1,2,5) (7,2)
%e (4,2,1) (1,3,4) (8,1)
%e (4,3,1) (1,2,6)
%e (5,2,1) (1,3,5)
%e (2,3,4)
%e (4,3,2)
%e (5,3,1)
%e (6,2,1)
%t Table[2*PartitionsQ[n]-1,{n,0,30}]
%Y Strict partitions are A000009.
%Y Unimodal compositions are A001523 (strict: A072706).
%Y Strict compositions are A032020.
%Y The non-strict version appears to be A329398.
%Y Partitions with incr. or decr. run-lengths are A332745 (strict: A333190).
%Y Compositions with incr. or decr. run-lengths are A332835 (strict: A333191).
%Y The complement is counted by A333149 (non-strict: A332834).
%Y Cf. A059204, A072705, A072707, A115981, A332285, A332578, A332746, A332831, A332833, A332874, A333150.
%K nonn
%O 0,4
%A _Gus Wiseman_, May 16 2020