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A355394
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Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.
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11
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1, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 18, 25, 30, 38, 47, 59, 74, 90, 112, 136, 171, 203, 253, 299, 372, 438, 536, 631, 767, 900, 1085, 1271, 1521, 1774, 2112, 2463, 2910, 3389, 3977, 4627, 5408, 6276, 7304, 8459, 9808, 11338, 13099, 15112, 17404, 20044, 23018, 26450, 30299, 34746, 39711, 45452, 51832
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OFFSET
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0,6
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COMMENTS
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These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).
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LINKS
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FORMULA
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EXAMPLE
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The a(0) = 1 through a(9) = 11 partitions:
() . . (21) (211) (32) (321) (43) (332) (54)
(221) (2211) (322) (3221) (432)
(2111) (21111) (2221) (22211) (3222)
(3211) (32111) (3321)
(22111) (221111) (22221)
(211111) (2111111) (32211)
(222111)
(321111)
(2211111)
(21111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Function[ptn, !Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]]], {n, 0, 30}]
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CROSSREFS
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These partitions are ranked by A356736.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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