A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

1, 1, 2, 2, 4, 5, 7, 10, 14, 18, 24, 31, 41, 53, 70, 87, 112, 140, 178, 221, 277, 344, 428, 526, 648, 792, 971, 1180, 1436, 1738, 2103, 2533, 3049, 3660, 4387, 5242, 6259, 7450, 8860, 10511, 12453, 14723, 17387, 20489, 24121, 28343, 33269, 38982, 45632, 53327

Offset: 0

Author

Gus Wiseman, Jan 18 2022

Keywords

• nonn

Comments

The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).

Examples

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (32)     (33)      (43)
                    (31)    (41)     (51)      (52)
                    (1111)  (311)    (222)     (61)
                            (11111)  (411)     (322)
                                     (3111)    (331)
                                     (111111)  (511)
                                               (4111)
                                               (31111)
                                               (1111111)
		

Links

• Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Crossrefs

The version with quotients >= 2 is A000929, sets A018819.

<= 2 is A342094, ranked by A342191.

< 2 is A342096, sets A045690, strict A342097.

> 2 is A342098, sets A040039.

The sets version (subsets of prescribed maximum) is A045691.

These partitions are ranked by A350838.

The strict case is A350840.

A version for differences is A350842, strict A350844.

The complement is counted by A350846, ranked by A350845.

A000041 = integer partitions.

A116931 = partitions with no successions, ranked by A319630.

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

A323092 = double-free partitions, ranked by A320340.

Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]