%I #20 Feb 12 2024 16:32:02
%S 0,1,1,1,2,2,3,4,6,6,9,11,13,17,20,24,30,36,41,52,60,71,84,100,114,
%T 137,158,183,214,248,283,330,379,432,499,570,648,742,846,955,1092,
%U 1234,1395,1580,1786,2005,2270,2548,2861,3216,3610,4032,4526,5055,5642,6304,7031,7820,8720,9694
%N Number of strict integer partitions of n with at least one neighborless part.
%C A part x is neighborless if neither x - 1 nor x + 1 are parts.
%H Alois P. Heinz, <a href="/A356607/b356607.txt">Table of n, a(n) for n = 0..5000</a> (first 101 terms from Lucas A. Brown)
%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A356607.py">A356607.py</a>
%e The a(0) = 0 through a(9) = 6 partitions:
%e . (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (31) (41) (42) (52) (53) (63)
%e (51) (61) (62) (72)
%e (421) (71) (81)
%e (431) (531)
%e (521) (621)
%t Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]
%Y This is the strict case of A356235 and A356236.
%Y The complement is counted by A356606, non-strict A355393 and A355394.
%Y A000041 counts integer partitions, strict A000009.
%Y A000837 counts relatively prime partitions, ranked by A289509.
%Y A007690 counts partitions with no singletons, complement A183558.
%Y Cf. A073492, A137921, A325160, A328171, A328172, A328187, A328220, A328221.
%K nonn
%O 0,5
%A _Gus Wiseman_, Aug 26 2022
%E a(31)-a(59) from _Lucas A. Brown_, Sep 09 2022