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A320388
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Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing.
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3
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1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 14, 16, 19, 18, 21, 25, 23, 26, 31, 29, 33, 38, 36, 40, 46, 44, 49, 56, 53, 58, 66, 64, 70, 77, 76, 82, 92, 89, 96, 106, 104, 113, 123, 120, 130, 142, 141, 149, 162, 160, 172, 186, 184, 195, 211, 210, 223, 238
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OFFSET
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0,4
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COMMENTS
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Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) > p(k) - p(k-1) for all k >= 3.
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LINKS
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Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..100 from Seiichi Manyama)
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EXAMPLE
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There are a(17) = 15 such partitions of 17:
01: [17]
02: [1, 16]
03: [2, 15]
04: [3, 14]
05: [4, 13]
06: [5, 12]
07: [6, 11]
08: [7, 10]
09: [1, 6, 10]
10: [8, 9]
11: [1, 7, 9]
12: [2, 6, 9]
13: [2, 7, 8]
14: [3, 6, 8]
15: [4, 6, 7]
There are a(18) = 14 such partitions of 18:
01: [18]
02: [1, 17]
03: [2, 16]
04: [3, 15]
05: [4, 14]
06: [5, 13]
07: [6, 12]
08: [7, 11]
09: [8, 10]
10: [1, 7, 10]
11: [1, 8, 9]
12: [2, 7, 9]
13: [3, 7, 8]
14: [1, 4, 6, 7]
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PROG
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(Ruby)
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
}
cnt
end
def A320388(n)
(0..n).map{|i| f(i)}
end
p A320388(50)
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CROSSREFS
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Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320387.
Cf. A081489.
Sequence in context: A241951 A084630 A325393 * A264396 A007360 A029144
Adjacent sequences: A320385 A320386 A320387 * A320389 A320390 A320391
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KEYWORD
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nonn
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AUTHOR
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Seiichi Manyama, Oct 12 2018
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STATUS
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approved
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