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

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, 108, 110, 135, 152, 166, 191, 223, 237, 275, 306, 345, 380, 429, 472, 537, 588, 650, 721, 808, 902, 972, 1083, 1205, 1316, 1450, 1617, 1742, 1919, 2130, 2312, 2531

Offset: 0

Author

David S. Newman, Dec 30 2010

Keywords

• nonn

Comments

a(0) = 1 as is common practice with partitions.

Examples

			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.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400
• Index entries related to non-averaging sequences

Crossrefs

Cf. A003407, A238569, A238571, A238687.

Programs

Mathematica

a[n_] := If[n == 0, 1, Select[IntegerPartitions[n],
     With[{u = Union[#]}, Length[#] == Length[u] &&
     SequencePosition[u, {b_, _, c_, _, d_} /;
     b-c == c-d, 1] == {}]&] // Length];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Aug 20 2021 *)

Sage

has_arith_prog = lambda x, size: any(len(set(differences(c))) <= 1 for c in Combinations(x,size))
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]