A164934 Number of different ways to select 3 disjoint subsets from {1..n} with equal element sum.

0, 0, 0, 0, 1, 3, 8, 22, 63, 157, 502, 1562, 4688, 15533, 50953, 165054, 562376, 1911007, 6467143, 22447463, 78021923, 271410289, 957082911, 3384587525, 11998851674, 42876440587, 153684701645, 552421854011, 1995875594696, 7231871165277, 26274832876337

Offset

1,6

Comments

a(5) = 1, because {1,4}, {2,3}, {5} are disjoint subsets of {1..5} with element sum 5.

a(6) = 3: {1,4}, {2,3}, {5} have element sum 5, {1,5}, {2,4}, {6} have element sum 6, and {1,6}, {2,5}, {3,4} have element sum 7.

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..104 (first 65 terms from Alois P. Heinz)

Formula

Conjecture: a(n) ~ 4^n / (Pi * sqrt(3) * n^3). - Vaclav Kotesovec, Oct 16 2014

Maple

b:= proc(n, k, i) option remember; local m;
      m:= i*(i+1)/2;
      if k>n then b(k, n, i)
    elif k>=0 and n+k>m or k<0 and n-2*k>m then 0
    elif [n, k, i] = [0, 0, 0] then 1
    else b(n, k, i-1)+b(n+i, k+i, i-1)+b(n-i, k, i-1)+b(n, k-i, i-1)
      fi
    end:
a:= proc(n) option remember;
      `if`(n>2, b(n, n, n-1)/2+ a(n-1), 0)
    end:
seq(a(n), n=1..20);

Mathematica

b[n_, k_, i_] := b[n, k, i] = Module[{m = i*(i+1)/2}, Which[k>n , b[k, n, i], k >= 0 && n+k>m || k<0 && n-2*k > m, 0, {n, k, i} == {0, 0, 0}, 1, True, b[n, k, i-1] + b[n+i, k+i, i-1] + b[n-i, k, i-1] + b[n, k-i, i-1]]]; a[n_] := a[n] = If[n>2, b[n, n, n-1]/2 + a[n-1], 0]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Crossrefs

Column k=3 of A196231.

Cf. A161943, A164949, A232534.

Sequence in context: A298260 A336990 A317997 * A047926 A192681 A339288

Adjacent sequences: A164931 A164932 A164933 * A164935 A164936 A164937

Keyword

nonn

Author

Alois P. Heinz, Aug 31 2009

Status

approved