A356607 Number of strict integer partitions of n with at least one neighborless part.

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, 137, 158, 183, 214, 248, 283, 330, 379, 432, 499, 570, 648, 742, 846, 955, 1092, 1234, 1395, 1580, 1786, 2005, 2270, 2548, 2861, 3216, 3610, 4032, 4526, 5055, 5642, 6304, 7031, 7820, 8720, 9694

Offset

0,5

Comments

A part x is neighborless if neither x - 1 nor x + 1 are parts.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)

Lucas A. Brown, A356607.py

Example

The a(0) = 0 through a(9) = 6 partitions:
  .  (1)  (2)  (3)  (4)   (5)   (6)   (7)    (8)    (9)
                    (31)  (41)  (42)  (52)   (53)   (63)
                                (51)  (61)   (62)   (72)
                                      (421)  (71)   (81)
                                             (431)  (531)
                                             (521)  (621)

Mathematica

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}]

Crossrefs

This is the strict case of A356235 and A356236.

The complement is counted by A356606, non-strict A355393 and A355394.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A073492, A137921, A325160, A328171, A328172, A328187, A328220, A328221.

Sequence in context: A046934 A093594 A008806 * A366843 A370805 A238860

Adjacent sequences: A356604 A356605 A356606 * A356608 A356609 A356610

Keyword

nonn

Author

Gus Wiseman, Aug 26 2022

Extensions

a(31)-a(59) from Lucas A. Brown, Sep 09 2022

Status

approved