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Number of strict integer partitions of n with no adjacent parts of quotient 2.
9

%I #10 Jan 27 2022 20:46:22

%S 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 106,122,142,166,190,216,249,287,325,371,420,479,543,617,695,784,888,

%U 1000,1126,1266,1420,1594,1792,2008,2247,2514,2809,3135,3496,3891,4332

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

%e The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):

%e 1 2 3 4 5 6 7 8 9 A B C D

%e 31 32 51 43 53 54 64 65 75 76

%e 41 52 62 72 73 74 93 85

%e 61 71 81 82 83 A2 94

%e 431 432 91 92 B1 A3

%e 531 532 A1 543 B2

%e 541 641 651 C1

%e 731 732 643

%e 741 652

%e 831 751

%e 832

%e 931

%e 5431

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

%Y The version for subsets of prescribed maximum is A045691.

%Y The double-free case is A120641.

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

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

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

%Y Versions for prescribed quotients:

%Y = 2: A154402, sets A001511.

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

%Y >= 2: A000929, sets A018819.

%Y <= 2: A342095, non-strict A342094.

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

%Y > 2: A342098, sets A040039.

%Y A000041 = integer partitions.

%Y A000045 = sets containing n with all differences > 2.

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

%Y A116931 = partitions with no successions, ranked by A319630.

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

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

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

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Jan 20 2022