OFFSET
0,3
COMMENTS
Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).
LINKS
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
EXAMPLE
The a(1) = 1 through a(8) = 8 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (11111) (33) (1111111) (44)
(211) (222) (422)
(1111) (3111) (2222)
(111111) (4211)
(41111)
(221111)
(11111111)
For example, the partition (3,2,2,2,1,1,1) has trajectory: (1,1,1,2,2,2,3) -> (3,3,6) -> (6,6) -> (12), so is counted under a(12).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&, #, !UnsameQ@@#&]]<=1&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 26 2022
STATUS
approved