A320388 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing.

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

Offset: 0

Author

Seiichi Manyama, Oct 12 2018

Keywords

• nonn

Comments

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.

Examples

			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]
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..100 from Seiichi Manyama)

Crossrefs

Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320387.

Cf. A081489.

Programs

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)