OFFSET
0,5
COMMENTS
A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).
EXAMPLE
The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(31) (41) (42) (52) (53)
(311) (51) (61) (62)
(411) (331) (71)
(3111) (421) (422)
(511) (431)
(4111) (521)
(31111) (611)
(4211)
(5111)
(41111)
(311111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Min@@Length/@Split[Reverse[#], #1>=#2-1&]==1&]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 23 2022
STATUS
approved