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%I #52 Feb 20 2023 06:14:51
%S 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,
%T 61,72,77,82,96,102,108,124,133,141,160,171,180,203,218,230,256,273,
%U 289,320,342,361,395,423,447,486,520,548,594,635,669,721,769,811,871,928,978,1044,1114
%N Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.
%C 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.
%H Alois P. Heinz, <a href="/A179254/b179254.txt">Table of n, a(n) for n = 0..1000</a> (first 225 terms from Joerg Arndt)
%e There are a(17) = 21 such partitions of 17:
%e 01: [ 1 2 4 10 ]
%e 02: [ 1 2 5 9 ]
%e 03: [ 1 2 14 ]
%e 04: [ 1 3 13 ]
%e 05: [ 1 4 12 ]
%e 06: [ 1 5 11 ]
%e 07: [ 1 16 ]
%e 08: [ 2 3 12 ]
%e 09: [ 2 4 11 ]
%e 10: [ 2 5 10 ]
%e 11: [ 2 15 ]
%e 12: [ 3 4 10 ]
%e 13: [ 3 5 9 ]
%e 14: [ 3 14 ]
%e 15: [ 4 5 8 ]
%e 16: [ 4 13 ]
%e 17: [ 5 12 ]
%e 18: [ 6 11 ]
%e 19: [ 7 10 ]
%e 20: [ 8 9 ]
%e 21: [ 17 ]
%e - _Joerg Arndt_, Mar 31 2014
%o (Sage)
%o def A179254(n):
%o has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
%o allowed = lambda x: len(x) < 3 or has_increasing_diffs(x)
%o return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
%o # _D. S. McNeil_, Jan 06 2011
%o (Ruby)
%o def partition(n, min, max)
%o return [[]] if n == 0
%o [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
%o end
%o def f(n)
%o return 1 if n == 0
%o cnt = 0
%o partition(n, 1, n).each{|ary|
%o ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
%o }
%o cnt
%o end
%o def A179254(n)
%o (0..n).map{|i| f(i)}
%o end
%o p A179254(50) # _Seiichi Manyama_, Oct 12 2018
%Y Cf. A007294, A179255 (nondecreasing differences), A179269, A320382, A320385.
%Y Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences).
%K nonn
%O 0,4
%A _Joerg Arndt_, Jan 05 2011
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