OFFSET
0,4
COMMENTS
These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
EXAMPLE
The a(0) = 1 through a(8) = 10 partitions:
() . (11) (21) (22) (32) (33) (43) (44)
(111) (211) (221) (222) (322) (332)
(1111) (2111) (321) (2221) (2222)
(11111) (2211) (3211) (3221)
(21111) (22111) (3311)
(111111) (211111) (22211)
(1111111) (32111)
(221111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Function[ptn, !Or@@Table[Count[ptn, x]==1&&!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 26 2022
STATUS
approved