A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 7, 8, 10, 13, 17, 19, 22, 25, 30, 35, 43, 52, 60, 70, 81, 93, 106, 122, 142, 166, 190, 216, 249, 287, 325, 371, 420, 479, 543, 617, 695, 784, 888, 1000, 1126, 1266, 1420, 1594, 1792, 2008, 2247, 2514, 2809, 3135, 3496, 3891, 4332

Offset: 0

Author

Gus Wiseman, Jan 20 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
  1   2   3   4    5    6    7    8     9     A     B     C     D
              31   32   51   43   53    54    64    65    75    76
                   41        52   62    72    73    74    93    85
                             61   71    81    82    83    A2    94
                                  431   432   91    92    B1    A3
                                        531   532   A1    543   B2
                                              541   641   651   C1
                                                    731   732   643
                                                          741   652
                                                          831   751
                                                                832
                                                                931
                                                                5431
		

Crossrefs

The version for subsets of prescribed maximum is A045691.

The double-free case is A120641.

The non-strict case is A350837, ranked by A350838.

An additive version (differences) is A350844, non-strict A350842.

The non-strict complement is counted by A350846, ranked by A350845.

Versions for prescribed quotients:

= 2: A154402, sets A001511.

!= 2: A350840 (this sequence), sets A045691.

>= 2: A000929, sets A018819.

<= 2: A342095, non-strict A342094.

< 2: A342097, non-strict A342096, sets A045690.

> 2: A342098, sets A040039.

A000041 = integer partitions.

A000045 = sets containing n with all differences > 2.

A003114 = strict partitions with no successions, ranked by A325160.

A116931 = partitions with no successions, ranked by A319630.

A116932 = partitions with differences != 1 or 2, strict A025157.

A323092 = double-free integer partitions, ranked by A320340.

A350839 = partitions with gaps and conjugate gaps, ranked by A350841.

Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]