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A178932 Partitions into distinct parts where no subset of the summands is an arithmetic progression (of length 3 or more).

%I A178932 #21 Aug 20 2021 09:01:49
%S A178932 1,1,1,2,2,3,3,5,6,6,9,11,11,15,19,18,26,29,32,38,48,47,62,68,79,89,
%T A178932 108,110,135,152,166,191,223,237,275,306,345,380,429,472,537,588,650,
%U A178932 721,808,902,972,1083,1205,1316,1450,1617,1742,1919,2130,2312,2531
%N A178932 Partitions into distinct parts where no subset of the summands is an arithmetic progression (of length 3 or more).
%C A178932 a(0) = 1 as is common practice with partitions.
%H A178932 Fausto A. C. Cariboni, <a href="/A178932/b178932.txt">Table of n, a(n) for n = 0..400</a>
%H A178932 <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</a>
%e A178932 There are 4 partitions of 6 into distinct parts, 6, 5+1, 4+2, and 3+2+1.  Since 3+2+1 contains the arithmetic progression 3,2,1, it won't be counted here.  Thus a(6)=3.
%t A178932 a[n_] := If[n == 0, 1, Select[IntegerPartitions[n],
%t A178932      With[{u = Union[#]}, Length[#] == Length[u] &&
%t A178932      SequencePosition[u, {b_, ___, c_, ___, d_} /;
%t A178932      b-c == c-d, 1] == {}]&] // Length];
%t A178932 Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 60}] (* _Jean-François Alcover_, Aug 20 2021 *)
%o A178932 (Sage) has_arith_prog = lambda x, size: any(len(set(differences(c))) <= 1 for c in Combinations(x,size))
%o A178932 A178932 = lambda n: Partitions(n,max_slope=-1).filter(lambda p: not has_arith_prog(sorted(p),3)).cardinality() # [_D. S. McNeil_, Dec 31 2010]
%Y A178932 Cf. A003407, A238569, A238571, A238687.
%K A178932 nonn
%O A178932 0,4
%A A178932 _David S. Newman_, Dec 30 2010