A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

1, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 18, 25, 30, 38, 47, 59, 74, 90, 112, 136, 171, 203, 253, 299, 372, 438, 536, 631, 767, 900, 1085, 1271, 1521, 1774, 2112, 2463, 2910, 3389, 3977, 4627, 5408, 6276, 7304, 8459, 9808, 11338, 13099, 15112, 17404, 20044, 23018, 26450, 30299, 34746, 39711, 45452, 51832

Offset

0,6

Comments

These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).

Links

Lucas A. Brown, Table of n, a(n) for n = 0..100

Lucas A. Brown, A355394.py

Formula

a(n) = A000041(n) - A356236(n).

Example

The a(0) = 1 through a(9) = 11 partitions:
  ()  .  .  (21)  (211)  (32)    (321)    (43)      (332)      (54)
                         (221)   (2211)   (322)     (3221)     (432)
                         (2111)  (21111)  (2221)    (22211)    (3222)
                                          (3211)    (32111)    (3321)
                                          (22111)   (221111)   (22221)
                                          (211111)  (2111111)  (32211)
                                                               (222111)
                                                               (321111)
                                                               (2211111)
                                                               (21111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Function[ptn, !Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]

Crossrefs

The singleton case is A355393, complement A356235.

The complement is counted by A356236, ranked by A356734.

The strict case is A356606, complement A356607.

These partitions are ranked by A356736.

A000041 counts integer partitions, strict A000009.

A000837 counts relatively prime partitions, ranked by A289509.

A007690 counts partitions with no singletons, complement A183558.

Cf. A066312, A073491, A077855, A328171, A328172, A328187, A328221, A356233, A356237.

Sequence in context: A026925 A343481 A237665 * A088528 A363241 A220153

Adjacent sequences: A355391 A355392 A355393 * A355395 A355396 A355397

Keyword

nonn

Author

Gus Wiseman, Aug 26 2022

Extensions

a(31)-a(59) from Lucas A. Brown, Sep 04 2022

Status

approved