OFFSET
0,8
COMMENTS
Also the number of non-weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.
EXAMPLE
The a(6) = 1 through a(10) = 13 partitions:
(321) (421) (431) (432) (532)
(3211) (521) (531) (541)
(4211) (621) (631)
(32111) (3321) (721)
(4311) (4321)
(5211) (5311)
(42111) (6211)
(321111) (32221)
(33211)
(43111)
(52111)
(421111)
(3211111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&!And@@EvenQ/@Take[Length/@Split[#], {2, -2}]&]], {n, 0, 30}]
CROSSREFS
The complement is counted by A349060.
These partitions are ranked by A349794.
The strong case is A349801.
A000041 counts integer partitions.
A003242 counts Carlitz (anti-run) compositions.
A096441 counts weakly alternating 0-appended partitions.
A349052 counts weakly alternating compositions.
A349056 counts weakly alternating permutations of prime indices.
A349798 counts weakly but not strongly alternating perms of prime indices.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 06 2021
STATUS
approved