A196723 Number of subsets of {1..n} (including empty set) such that the pairwise sums of distinct elements are all distinct.

1, 2, 4, 8, 15, 28, 50, 86, 143, 236, 376, 594, 913, 1380, 2048, 3016, 4367, 6302, 8974, 12670, 17685, 24580, 33738, 46072, 62367, 83990, 112342, 149734, 198153, 261562, 343210, 448694, 583445, 756846, 976086, 1255658, 1607831, 2053186, 2610560, 3312040, 4183689

Offset: 0

Author

Alois P. Heinz, Oct 06 2011

Keywords

• nonn

Comments

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum is A143823(n).

Examples

			a(4) = 15: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..110

Crossrefs

Cf. A143823, A196719, A196720, A196721, A196722, A196724.

The subset case is A196723 (this sequence).

The maximal case is A325878.

The integer partition case is A325857.

The strict integer partition case is A325877.

Heinz numbers of the counterexamples are given by A325991.

Cf. A108917, A325858, A325862, A325863, A325864.

Programs

Maple

b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       sn[i]+sn[j], j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..20);

Mathematica

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n<1, 1, b[n-1, s] + If[m*(m+1)/2 == Length[ Union[ Flatten[ Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i+1, m+1}]]]], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2017, translated from Maple *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#,{2}]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

Extensions

Edited by Gus Wiseman, Jun 03 2019