A143824 Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset

0,3

Comments

See A143823 for the number of subsets of {1, 2, ..., n} with the required property.

See A003022 (and A227590) for the values of n such that a(n+1) > a(n). - Boris Bukh, Jul 28 2013

Can be formulated as an integer linear program: maximize sum {i = 1 to n} z[i] subject to z[i] + z[j] - 1 <= y[i,j] for all i < j, sum {i = 1 to n - d} y[i,i+d] <= 1 for d = 1 to n - 1, z[i] in {0,1} for all i, y[i,j] in {0,1} for all i < j. - Rob Pratt, Feb 09 2010

If the zeroth term is removed, the run-lengths are A270813 with 1 prepended. - Gus Wiseman, Jun 07 2019

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..500

Formula

For n > 1, a(n) = A325678(n - 1) + 1. - Gus Wiseman, Jun 07 2019

Example

For n = 4, {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 and no larger set has the required property; so a(4) = 3.
From Gus Wiseman, Jun 07 2019: (Start)
Examples of subsets realizing each largest size are:
   0: {}
   1: {1}
   2: {1,2}
   3: {2,3}
   4: {1,3,4}
   5: {2,4,5}
   6: {3,5,6}
   7: {1,3,6,7}
   8: {2,4,7,8}
   9: {3,5,8,9}
  10: {4,6,9,10}
  11: {5,7,10,11}
  12: {1,4,5,10,12}
  13: {2,5,6,11,13}
  14: {3,6,7,12,14}
  15: {4,7,8,13,15}
(End)

Mathematica

Table[Length[Last[Select[Subsets[Range[n]], UnsameQ@@Subtract@@@Subsets[#, {2}]&]]], {n, 0, 15}] (* Gus Wiseman, Jun 07 2019 *)

Crossrefs

Cf. A143823, A003002, A003022, A227590.

Cf. A108917, A169942, A270813, A325676, A325677, A325678, A325683, A325687.

Sequence in context: A238965 A036042 A162988 * A182009 A034463 A259899

Adjacent sequences: A143821 A143822 A143823 * A143825 A143826 A143827

Keyword

nonn

Author

John W. Layman, Sep 02 2008

Extensions

More terms from Rob Pratt, Feb 09 2010

a(41)-a(60) from Alois P. Heinz, Sep 14 2011

More terms and b-file from N. J. A. Sloane, Apr 08 2016 using data from A003022.

Status

approved