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

%I A350840 #10 Jan 27 2022 20:46:22
%S A350840 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,
%T A350840 106,122,142,166,190,216,249,287,325,371,420,479,543,617,695,784,888,
%U A350840 1000,1126,1266,1420,1594,1792,2008,2247,2514,2809,3135,3496,3891,4332
%N A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.
%e A350840 The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
%e A350840   1   2   3   4    5    6    7    8     9     A     B     C     D
%e A350840               31   32   51   43   53    54    64    65    75    76
%e A350840                    41        52   62    72    73    74    93    85
%e A350840                              61   71    81    82    83    A2    94
%e A350840                                   431   432   91    92    B1    A3
%e A350840                                         531   532   A1    543   B2
%e A350840                                               541   641   651   C1
%e A350840                                                     731   732   643
%e A350840                                                           741   652
%e A350840                                                           831   751
%e A350840                                                                 832
%e A350840                                                                 931
%e A350840                                                                 5431
%t A350840 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]
%Y A350840 The version for subsets of prescribed maximum is A045691.
%Y A350840 The double-free case is A120641.
%Y A350840 The non-strict case is A350837, ranked by A350838.
%Y A350840 An additive version (differences) is A350844, non-strict A350842.
%Y A350840 The non-strict complement is counted by A350846, ranked by A350845.
%Y A350840 Versions for prescribed quotients:
%Y A350840   = 2:  A154402, sets A001511.
%Y A350840   != 2: A350840 (this sequence), sets A045691.
%Y A350840   >= 2: A000929, sets A018819.
%Y A350840   <= 2: A342095, non-strict A342094.
%Y A350840   < 2:  A342097, non-strict A342096, sets A045690.
%Y A350840   > 2:  A342098, sets A040039.
%Y A350840 A000041 = integer partitions.
%Y A350840 A000045 = sets containing n with all differences > 2.
%Y A350840 A003114 = strict partitions with no successions, ranked by A325160.
%Y A350840 A116931 = partitions with no successions, ranked by A319630.
%Y A350840 A116932 = partitions with differences != 1 or 2, strict A025157.
%Y A350840 A323092 = double-free integer partitions, ranked by A320340.
%Y A350840 A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
%Y A350840 Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.
%K A350840 nonn
%O A350840 0,5
%A A350840 _Gus Wiseman_, Jan 20 2022