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A232466 Number of dependent sets with largest element n. 2
0, 0, 1, 2, 4, 10, 20, 44, 93, 198, 414, 864, 1788, 3687, 7541, 15382, 31200, 63191, 127482, 256857, 516404, 1037104, 2080357, 4170283, 8354078 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Let S be a set of positive integers. If S can be divided into two subsets which have equal sums, then S is said to be a dependent set.

REFERENCES

J. Bourgain, Λ_p-sets in analysis: results, problems and related aspects. Handbook of the geometry of Banach spaces, Vol. I,195-232, North-Holland, Amsterdam, 2001.

LINKS

Table of n, a(n) for n=1..25.

EXAMPLE

Here are the dependent sets counted by a(6): {1,5,6}, {2,4,6}, {1,2,3,6}, {1,2,5,6}, {1,3,4,6}, {2,3,5,6}, {3,4,5,6}, {1,2,3,4,6}, {1,2,4,5,6}, {2,3,4,5,6}.

MAPLE

b:= proc(n, i) option remember; `if`(i<1, `if`(n=0, {0}, {}),

      `if`(i*(i+1)/2<n, {}, b(n, i-1) union map(p-> p+x^i,

       b(n+i, i-1) union b(abs(n-i), i-1))))

    end:

a:= n-> nops(b(n, n-1)):

seq(a(n), n=1..15);  # Alois P. Heinz, Nov 24 2013

MATHEMATICA

b[n_, i_] := b[n, i] = If[i<1, If[n == 0, {0}, {}], If[i*(i+1)/2 < n, {}, b[n, i-1] ~Union~ Map[Function[p, p+x^i], b[n+i, i-1] ~Union~ b[Abs[n-i], i-1]]]]; a[n_] := Length[b[n, n-1]]; Table[Print[a[n]]; a[n], {n, 1, 24}] (* Jean-François Alcover, Mar 04 2014, after Alois P. Heinz *)

PROG

(PARI) dep(S, k=0)=if(#S<2, return(if(#S, S[1], 0)==k)); my(T=S[1..#S-1]); dep(T, abs(k-S[#S]))||dep(T, k+S[#S])

a(n)=my(S=[1..n-1]); sum(i=1, 2^(n-1)-1, dep(vecextract(S, i), n)) \\ Charles R Greathouse IV, Nov 25 2013

CROSSREFS

Cf. A161943, A232534.

Column k=2 of A248112.

Sequence in context: A121880 A094536 A297183 * A003407 A151523 A265264

Adjacent sequences:  A232463 A232464 A232465 * A232467 A232468 A232469

KEYWORD

nonn,more

AUTHOR

David S. Newman, Nov 24 2013

EXTENSIONS

a(9)-a(24) from Alois P. Heinz, Nov 24 2013

a(25) from Alois P. Heinz, Sep 30 2014

STATUS

approved

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Last modified February 18 15:08 EST 2018. Contains 299324 sequences. (Running on oeis4.)