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 A179254 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing. 13
 1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 14, 15, 19, 21, 22, 28, 30, 32, 39, 42, 44, 54, 58, 61, 72, 77, 82, 96, 102, 108, 124, 133, 141, 160, 171, 180, 203, 218, 230, 256, 273, 289, 320, 342, 361, 395, 423, 447, 486, 520, 548, 594, 635, 669, 721, 769, 811, 871, 928, 978, 1044, 1114 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 225 terms from Joerg Arndt) EXAMPLE There are a(17) = 21 such partitions of 17: 01:  [ 1 2 4 10 ] 02:  [ 1 2 5 9 ] 03:  [ 1 2 14 ] 04:  [ 1 3 13 ] 05:  [ 1 4 12 ] 06:  [ 1 5 11 ] 07:  [ 1 16 ] 08:  [ 2 3 12 ] 09:  [ 2 4 11 ] 10:  [ 2 5 10 ] 11:  [ 2 15 ] 12:  [ 3 4 10 ] 13:  [ 3 5 9 ] 14:  [ 3 14 ] 15:  [ 4 5 8 ] 16:  [ 4 13 ] 17:  [ 5 12 ] 18:  [ 6 11 ] 19:  [ 7 10 ] 20:  [ 8 9 ] 21:  [ 17 ] - Joerg Arndt, Mar 31 2014 PROG (Sage) def A179254(n):     has_increasing_diffs = lambda x: min(differences(x, 2)) >= 1     allowed = lambda x: len(x) < 3 or has_increasing_diffs(x)     return len([x for x in Partitions(n, max_slope=-1) if allowed(x[::-1])]) # D. S. McNeil, Jan 06 2011_ (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.reverse && ary0.uniq == ary0   }   cnt end def A179254(n)   (0..n).map{|i| f(i)} end p A179254(50) # Seiichi Manyama, Oct 12 2018 CROSSREFS Cf. A007294, A179255 (nondecreasing differences), A179269, A320382, A320385. Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences). Sequence in context: A240542 A342516 A325391 * A304430 A086609 A341140 Adjacent sequences:  A179251 A179252 A179253 * A179255 A179256 A179257 KEYWORD nonn AUTHOR Joerg Arndt, Jan 05 2011 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)