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A143823
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Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.
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67
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1, 2, 4, 7, 13, 22, 36, 57, 91, 140, 216, 317, 463, 668, 962, 1359, 1919, 2666, 3694, 5035, 6845, 9188, 12366, 16417, 21787, 28708, 37722, 49083, 63921, 82640, 106722, 136675, 174895, 222558, 283108, 357727, 451575, 567536, 712856, 890405, 1112081, 1382416, 1717540
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OFFSET
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0,2
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COMMENTS
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See A143824 for sizes of the largest subsets of {1,2,...,n} with the desired property.
a(n) = A169947(n-1) + n + 1 for n>=2.
a(n) = A054578(n) + 1 for n>0. - Alois P. Heinz, Jan 17 2013
Also the number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum. - Gus Wiseman, Jun 07 2019
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LINKS
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Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100 (terms 0..60 from Alois P. Heinz, 61..81 from Vaclav Kotesovec)
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EXAMPLE
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{1,2,4} is a subset of {1,2,3,4}, with distinct differences 2-1=1, 4-1=3, 4-2=2 between pairs of elements, so {1,2,4} is counted as one of the 13 subsets of {1,2,3,4} with the desired property. Only 2^4-13=3 subsets of {1,2,3,4} do not have this property: {1,2,3}, {2,3,4}, {1,2,3,4}.
From Gus Wiseman, May 17 2019: (Start)
The a(0) = 1 through a(5) = 22 subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{1,2} {3} {3} {3}
{1,2} {4} {4}
{1,3} {1,2} {5}
{2,3} {1,3} {1,2}
{1,4} {1,3}
{2,3} {1,4}
{2,4} {1,5}
{3,4} {2,3}
{1,2,4} {2,4}
{1,3,4} {2,5}
{3,4}
{3,5}
{4,5}
{1,2,4}
{1,2,5}
{1,3,4}
{1,4,5}
{2,3,5}
{2,4,5}
(End)
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MAPLE
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b:= proc(n, s) local sn, m;
if n<1 then 1
else sn:= [s[], n];
m:= nops(sn);
`if`(m*(m-1)/2 = nops(({seq(seq(sn[i]-sn[j],
j=i+1..m), i=1..m-1)})), b(n-1, sn), 0) +b(n-1, s)
fi
end:
a:= proc(n) option remember;
b(n-1, [n]) +`if`(n=0, 0, a(n-1))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 14 2011
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MATHEMATICA
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b[n_, s_] := Module[{ sn, m}, If[n<1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m-1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m-1}, {j, i+1, m}] // Flatten // Union], b[n-1, sn], 0] + b[n-1, s]]]; a[n_] := a[n] = b[n - 1, {n}] + If[n == 0, 0, a[n-1]]; Table [a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)
Table[Length[Select[Subsets[Range[n]], UnsameQ@@Abs[Subtract@@@Subsets[#, {2}]]&]], {n, 0, 15}] (* Gus Wiseman, May 17 2019 *)
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CROSSREFS
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First differences are A308251.
Second differences are A169942.
The subset case is A143823 (this sequence).
The maximal case is A325879.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A143824, A054578, A169947.
Cf. A000079, A108917, A143824, A169942, A308251, A325676, A325677, A325679, A325683, A325860, A325864.
Sequence in context: A061255 A088111 A325864 * A119983 A151897 A192758
Adjacent sequences: A143820 A143821 A143822 * A143824 A143825 A143826
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KEYWORD
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nonn
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AUTHOR
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John W. Layman, Sep 02 2008
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EXTENSIONS
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a(21)-a(29) and connection to A169947 from Nathaniel Johnston, Nov 12 2010
Corrected a(21)-a(29) and more terms from Alois P. Heinz, Sep 14 2011
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STATUS
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approved
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