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

%I A350837 #12 Jan 25 2022 10:26:10
%S A350837 1,1,2,2,4,5,7,10,14,18,24,31,41,53,70,87,112,140,178,221,277,344,428,
%T A350837 526,648,792,971,1180,1436,1738,2103,2533,3049,3660,4387,5242,6259,
%U A350837 7450,8860,10511,12453,14723,17387,20489,24121,28343,33269,38982,45632,53327
%N A350837 Number of integer partitions of n with no adjacent parts of quotient 2.
%C A350837 The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).
%H A350837 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A350837 The a(1) = 1 through a(7) = 10 partitions:
%e A350837   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A350837        (11)  (111)  (22)    (32)     (33)      (43)
%e A350837                     (31)    (41)     (51)      (52)
%e A350837                     (1111)  (311)    (222)     (61)
%e A350837                             (11111)  (411)     (322)
%e A350837                                      (3111)    (331)
%e A350837                                      (111111)  (511)
%e A350837                                                (4111)
%e A350837                                                (31111)
%e A350837                                                (1111111)
%t A350837 Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]
%Y A350837 The version with quotients >= 2 is A000929, sets A018819.
%Y A350837                            <= 2 is A342094, ranked by A342191.
%Y A350837                            < 2 is A342096, sets A045690, strict A342097.
%Y A350837                            > 2 is A342098, sets A040039.
%Y A350837 The sets version (subsets of prescribed maximum) is A045691.
%Y A350837 These partitions are ranked by A350838.
%Y A350837 The strict case is A350840.
%Y A350837 A version for differences is A350842, strict A350844.
%Y A350837 The complement is counted by A350846, ranked by A350845.
%Y A350837 A000041 = integer partitions.
%Y A350837 A116931 = partitions with no successions, ranked by A319630.
%Y A350837 A116932 = partitions with differences != 1 or 2, strict A025157.
%Y A350837 A323092 = double-free partitions, ranked by A320340.
%Y A350837 Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.
%K A350837 nonn
%O A350837 0,3
%A A350837 _Gus Wiseman_, Jan 18 2022