A039799 For S a subset of [ n ] = {1,2,3,...n}, let B_S = {x+y : x,y in S, x<y}; then a(n) is maximal cardinality of B_S intersect B_{[ n ]-S}.

0, 0, 0, 1, 1, 2, 3, 6, 6, 9, 10, 12, 14, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119

Offset

1,6

Comments

An easy upper bound is 2n-7, since the sums in the intersection must come from {5,...,2n-3}. It is easy to see that 5 and 6 cannot both appear as sums. It appears that at most three sums can come from {5,...,9} and at most three from {2n-7,...,2n-3}. If this is true, then 2n-11 is an upper bound for n >= 9. - Rob Pratt, Jul 14 2015

Links

Rob Pratt, Table of n, a(n) for n = 1..100

Formula

Conjecture: a(n) = 2n-11 for n >= 14. - Rob Pratt, Jul 14 2015

Example

a(7) = 3 since we can divide [ 7 ] into S={1,5,6} and T={2,3,4,7} giving B_S={6,7,11} and B_T={5,6,7,9,10,11}, with intersection {6,7,11} of cardinality 3.

Maple

B:= proc(s) local l, i, j;
       l:= [s[]];
       {seq(seq(l[i]+l[j], i=1..j-1), j=2..nops(l))}
    end:
b:= proc(n, i, s)
      if i=0 then nops(B(s) intersect B({$1..n} minus s))
    else max(b(n, i-1, s), b(n, i-1, s union {i}))
      fi
    end;
a:= n-> b(n, n, {}):
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 28 2011

Mathematica

B[s_List] := B[s] = Module[{l=s}, Flatten @ Table[Table[l[[i]] + l[[j]], {i, 1, j-1}], {j, 2, Length[l]}]]; b[n_, i_, s_List] := b[n, i, s] = If[i == 0, Length[B[s] ~Intersection~ B[Range[n] ~Complement~ s]], Max[b[n, i-1, s], b[n, i-1, s ~Union~ {i}]]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, May 29 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A198516 A187326 A056907 * A185423 A144583 A328565

Adjacent sequences: A039796 A039797 A039798 * A039800 A039801 A039802

Keyword

hard,nonn

Author

Erich Friedman

Extensions

a(15)-a(23) from Alois P. Heinz, Feb 28 2011

a(24)-a(100) from Rob Pratt, Jul 14 2015

Status

approved